chap14 bk

Full text

(1)Probability Chapter Overview and Pacing. PACING (days) Regular Block. LESSON OBJECTIVES. Basic/ Average. Advanced. Basic/ Average. Advanced. Counting Outcomes (pp. 754–759) • Count outcomes using a tree diagram. • Count outcomes using the Fundamental Counting Principle. Follow-Up: Use finite graphs to determine whether a route is traceable.. optional. 1. optional. 0.5. Permutations and Combinations (pp. 760–767) • Determine probabilities using permutations. • Determine probabilities using combinations.. optional. 2. optional. 1. Probability of Compound Events (pp. 769–776) • Find the probability of two independent events or dependent events. • Find the probability of two mutually exclusive or inclusive events.. optional. 2. optional. 1. Probability Distributions (pp. 777–781) • Use random variables to compute probability. • Use probability distributions to solve real-world problems.. optional. 2. optional. 1. Probability Simulations (pp. 782–788) • Use theoretical and experimental probability to represent and solve problems involving uncertainty. • Perform probability simulations to model real-world situations involving uncertainty.. optional. 2. optional. 1. Study Guide and Practice Test (pp. 789–793) Standardized Test Practice (pp. 794–795). optional. 1. optional. 0.5. Chapter Assessment. optional. 1. optional. 0.5. TOTAL. Pacing suggestions for the entire year can be found on pages T20–T21.. 752A Chapter 14 Probability. 11. 5.5.

(2) Timesaving Tools ™. All-In-One Planner and Resource Center. Chapter Resource Manager. 831–832. 833–834. 835. 836. 837–838. 839–840. 841. 842. 875. 843–844. 845–846. 847. 848. 875, 877. 849–850. 851–852. 853. 854. 876. 855–856. 857–858. 859. 860. 876. Ap plic atio ns* Par Stu ent dy a Gu nd St ide u Wo dent rkb 5-M ook Tra inute nsp Che are nci ck es Int e Cha racti lkb ve oar d Alg ePA Plu SS: T s (l ess utoria ons l ). Ass ess me nt Pre req u Wo isite rkb Ski ook lls. Enr ich me nt. S and tudy Int Guid erv e ent ion (Sk Pra c ills and tice. Ave rag e). Rea di Ma ng to the ma Learn tics. CHAPTER 14 RESOURCE MASTERS. See pages T12–T13.. 107. 14-1. 14-1. 108. 14-2. 14-2. 109. 14-3. 14-3. GCS 50. 110. 14-4. 14-4. GCS 49, SC 28. 111. 14-5. 14-5. SC 27. 47–48, SM 109–114 55–56, 67–70, 99–100. 861–874, 878–880. Materials. 35. die, graphing calculator. 112. *Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual. ELL Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish. Chapter 14 Probability 752B.

(3) Mathematical Connections and Background Continuity of Instruction Prior Knowledge In Chapter 2, students were shown how to find the probability and odds of a simple event. While these techniques are great for use with dice, spinners, and board games, situations in the real world produce much more complex probability situations.. This Chapter In this chapter, students will go beyond simple probability. They will learn to find probabilities using the Fundamental Counting Principle, and by using permutations and combinations. Then students will move on to find probabilities of compound events, and to use probability distributions and simulations.. Future Connections Understanding probability can be an important decision-making tool. We are often asked to make decisions based on whether we think an event will occur. Using probability can take some of the guesswork out of these decisions.. Counting Outcomes Given a choice of three different sandwiches, four different side dishes, and five different soft drinks, how many lunch combinations are there? One can find out by making a tree diagram of all possible outcomes. In the first column, list all of the possibilities for one choice. After each of those possibilities, list all the possibilities of another choice. Repeat for all the possibilities for each choice that is to be made. The list of all possible outcomes is written in the last column. This is the sample space. Each item in the sample space is an event. Another method of counting outcomes uses the Fundamental Counting Principle. This principle states that given an event M that can occur m ways followed by an event N that occurs n ways, then the event M followed by event N can occur m n ways. In other words, number of choices number of choices number of choices…, for each choice that is to be made. A factorial is the product of a number and all the positive integers between that number and zero. Factorials can be used to determine the number of arrangements or orders for a set of data or events. They are also used to determine permutations and combinations that are studied in the next lesson.. Permutations and Combinations A permutation is an arrangement or listing in which order or placement is important. For example, if three students are being picked for class president, vicepresident, and secretary, then the order in which they are picked is important. Therefore, this is a permutation. Tree diagrams can be used to show permutations. A symbol for permutations is nPr , where n is the number of items to choose from and r is the number of items to be chosen. You find the number of permutations by n! (n r)!. using the formula . A combination is an arrangement in which order or placement is not important. For example when picking two pizza toppings from a list of eight, the order in which the toppings are picked is not important. So this is a combination. The symbol for a combination is nCr. The only way the formula for the number of combinations differs from the formula for the number of permutations is that the denominator is multiplied by r!. The n! (n r)!r!. formula is .. 752C. Chapter 14 Probability.

(4) Probability of Compound Events A simple event is a single event. A compound event is two or more simple events. Predicting the probability of rain on two separate days is an example of finding the probability of a compound event. If two events occur separately and the outcome of one does not affect the outcome of the other, then the events are independent. However, if the outcome of one event does affect the outcome of the other event, then the events are dependent. To find the compound probability of two independent events, find the product of the probabilities of each event: P(A) P(B). To find the probability of two dependent events, multiply the probability of the first event by the probability of the second event following the first event: P(A) P(B following A). Mutually exclusive events are events that cannot occur at the same time. An example is rolling an even number and an odd number on a die. An even and an odd number cannot be rolled at the same time with one die. To find the probability of mutually exclusive events, find the sum of the probabilities of the two events: P(A) + P(B). Inclusive events can occur at the same time. For example, rolling an odd number or a 5 on a die. Five is an odd number, so both can be rolled at the same time. Use the formula P(A) + P(B) – P(A and B) to find the probability of inclusive events.. Probability Distributions A random variable is a variable whose value is the numerical outcome of a random event. The probability of every possible value of the random variable X is called a probability distribution. A probability distribution can be represented in a table or in a probability histogram.. Probability Simulations The probabilities studied so far have been theoretical probabilities. These are probabilities that are determined mathematically and describe what should happen. Experimental probability is the probability of an event found by repeated experimentation. The experimentation is called a simulation. What should happen and what actually does happen may differ. Given few trials, experimental probability may differ greatly from the calculated theoretical probability. However, the more trials conducted, the more closely experimental probability will be to theoretical probability. For example, tossing a die 18 times may only result in one outcome of 1. However, theoretical 1 6. probability predicts a probability. If you toss the die 200 times, however, the experimental probability will 1 6. probably be very close to .. Chapter 14 Probability 752D.

(5) and Assessment. ASSESSMENT. INTERVENTION. Type. Student Edition. Teacher Resources. Ongoing. Prerequisite Skills, pp. 753, 758, 767, 776, 781 Practice Quiz 1, p. 767 Practice Quiz 2, p. 781. 5-Minute Check Transparencies Prerequisite Skills Workbook, pp. 47–48, 55–56, 67–70, 99–100 Quizzes, CRM pp. 875–876 Mid-Chapter Test, CRM p. 877 Study Guide and Intervention, CRM pp. 831–832, 837–838, 843–844, 849–850, 855–856. Mixed Review. pp. 758, 767, 776, 781, 788. Cumulative Review, CRM p. 878. Error Analysis. Find the Error, pp. 764, 773 Common Misconceptions, p. 760. Find the Error, TWE pp. 764, 773 Tips for New Teachers, TWE pp. 763, 779. Standardized Test Practice. pp. 758, 762, 764, 766, 776, 780, 787, 793, 794–795. TWE pp. 794–795 Standardized Test Practice, CRM pp. 879–880. Open-Ended Assessment. Writing in Math, pp. 757, 766, 776, 780, 787 Open Ended, pp. 756, 764, 772, 779, 785 Standardized Test, p. 795. Modeling: TWE pp. 758, 781, 788 Speaking: TWE p. 767 Writing: TWE p. 776 Open-Ended Assessment, CRM p. 873. Chapter Assessment. Study Guide, pp. 789–792 Practice Test, p. 793. Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 861–866 Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 867–872 Vocabulary Test/Review, CRM p. 874. Technology/Internet AlgePASS: Tutorial Plus www.algebra1.com/self_check_quiz www.algebra1.com/extra_examples. Standardized Test Practice CD-ROM www.algebra1.com/ standardized_test. TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes www.algebra1.com/ vocabulary_review www.algebra1.com/chapter_test. Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters. Additional Intervention Resources The Princeton Review’s Cracking the SAT & PSAT The Princeton Review’s Cracking the ACT ALEKS. TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment flexibility: • Worksheet Builder to make worksheet and tests • Student Module to take tests on screen (optional) • Management System to keep student records (optional) Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests.. 752E. Chapter 14 Probability.

(6) Reading and Writing in Mathematics Intervention Technology AlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum. Algebra 1 Lesson 14-3. AlgePASS Lesson 35 Integration: Introduction to Probability. ALEKS is an online mathematics learning system that adapts assessment and tutoring to the student’s needs. Subscribe at www.k12aleks.com.. Intervention at Home Parent and Student Study Guide Parents and students may work together to reinforce the concepts and skills of this chapter. (Workbook, pp. 107–112 or log on to www.algebra1.com/parent_student ) Log on for student study help. • For each lesson in the Student Edition, there are Extra Examples and Self-Check Quizzes. www.algebra1.com/extra_examples www.algebra1.com/self_check_quiz. • For chapter review, there is vocabulary review, test practice, and standardized test practice. www.algebra1.com/vocabulary_review www.algebra1.com/chapter_test www.algebra1.com/standardized_test. For more information on Intervention and Assessment, see pp. T8–T11.. Glencoe Algebra 1 provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition • Foldables Study Organizer, p. 753 • Concept Check questions require students to verbalize and write about what they have learned in the lesson. (pp. 756, 764, 772, 779, 785) • Reading Mathematics, p. 768 • Writing in Math questions in every lesson, pp. 757, 766, 776, 780, 787 • Reading Study Tip, pp. 771, 777 • WebQuest, pp. 766, 788 Teacher Wraparound Edition • Foldables Study Organizer, pp. 753, 789 • Study Notebook suggestions, pp. 756, 759, 764, 768, 772, 779, 785 • Modeling activities, pp. 758, 781, 788 • Speaking activities, p. 767 • Writing activities, p. 776 • ELL Resources, pp. 752, 757, 766, 768, 774, 780, 787, 789 Additional Resources • Vocabulary Builder worksheets require students to define and give examples for key vocabulary terms as they progress through the chapter. (Chapter 14 Resource Masters, pp. vii-viii) • Reading to Learn Mathematics master for each lesson (Chapter 14 Resource Masters, pp. 835, 841, 847, 853, 859) • Vocabulary PuzzleMaker software creates crossword, jumble, and word search puzzles using vocabulary lists that you can customize. • Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. • Reading and Writing in the Mathematics Classroom • WebQuest and Project Resources • Hot Words/Hot Topics Sections 2.1, 2.3, 2.4, 2.9, 4.5, 4.6. For more information on Reading and Writing in Mathematics, see pp. T6–T7. Chapter 14 Probability 752F.

(7) Notes Have students read over the list of objectives and make a list of any words with which they are not familiar.. Point out to students that this is only one of many reasons why each objective is important. Others are provided in the introduction to each lesson.. Probability • Lesson 14-1 Fundamental • Lesson 14-2 permutations • Lesson 14-3 events. • Lesson 14-4 • Lesson 14-5. Count outcomes using the Counting Principle. Determine probabilities using and combinations. Find probabilities of compound Use probability distributions. Use probability simulations.. Key Vocabulary • • • • •. permutation (p. 760) combination (p. 762) compound event (p. 769) theoretical probability (p. 782) experimental probability (p. 782). The United States Senate forms committees to focus on different issues. These committees are made up of senators from various states and political parties. There are many ways these committees could be formed. You will learn how to find the number of possible committees in Lesson 14-2.. Lesson 14-1 14-1 Follow-Up 14-2 14-3 14-4 14-5. NCTM Standards. Local Objectives. 1, 5, 6, 8, 9, 10 3, 5, 6, 7, 8, 9, 10 1, 5, 6, 8, 9, 10 1, 5, 6, 8, 9, 10 1, 5, 6, 8, 9, 10 1, 5, 6, 8, 9, 10. Key to NCTM Standards: 1=Number & Operations, 2=Algebra, 3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=Problem Solving, 7=Reasoning & Proof, 8=Communication, 9=Connections, 10=Representation 752. Chapter 14 Probability. 752 Chapter 14. Probability. Vocabulary Builder. ELL. The Key Vocabulary list introduces students to some of the main vocabulary terms included in this chapter. For a more thorough vocabulary list with pronunciations of new words, give students the Vocabulary Builder worksheets found on pages vii and viii of the Chapter 14 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they add these sheets to their study notebooks for future reference when studying for the Chapter 14 test..

(8) Prerequisite Skills To be successful in this chapter, you’ll need to master these skills and be able to apply them in problem-solving situations. Review these skills before beginning Chapter 14. For Lessons 14-2 through 14-5. This section provides a review of the basic concepts needed before beginning Chapter 14. Page references are included for additional student help. Additional review is provided in the Prerequisite Skills Workbook, pp. 47–48, 55–56, 67–70, 99–100.. Find Simple Probabilities. Determine the probability of each event if you randomly select a cube from a bag containing 6 red cubes, 3 blue cubes, 4 yellow cubes, and 1 green cube. (For review, see Lesson 2-6.). 3 2. P(blue) 14. 3 1. P(red) 7. 2 3. P(yellow) 7. 4 4. P(not red) 7. For Lesson 14-2 Find each product. (For review, see pages 800 and 801.) 5 6 5 4 3 3 5. 6. 12 11 22 5 4 5 7 4 7 13 4 1 8. 9. 32 32 256 52 52 52. 7 4 7 7. 20 19 95 56 24 84 10. 100 100 625. For Lesson 14-4. Write Decimals as Percents. Write each decimal as a percent. 11. 0.725 72.5%. Prerequisite Skills in the Getting Ready for the Next Lesson section at the end of each exercise set review a skill needed in the next lesson.. Multiply Fractions. For Lesson. (For review, see pages 804 and 805.). 12. 0.148 14.8%. 13. 0.4 40%. For Lesson 14-5. 14. 0.0168 1.68%. 14-2 14-4. Write Fractions as Percents. Write each fraction as a percent. Round to the nearest tenth. (For review, see pages 804 and 805.) 7 15. 87.5% 8. 33 16. 41.3% 80. 107 17. 85.6% 125. 14-5. 625 18. 61% 1024. Prerequisite Skill Simple Probability (p. 758) Expressing Fractions as Decimals (p. 776) Writing Fractions as Percents (p. 781). Make this Foldable to help you organize what you learn about 1 probability. Begin with a sheet of plain 8" by 11" paper. 2. Fold in Half. Fold Again in Fourths Fold the top to the bottom twice.. Fold in half lengthwise.. Cut. Label. Open. Cut along the second fold to make four tabs.. Label as shown.. Probability. ComOut- Permu- Combin- pound comes tations ations Events. As you read and study the chapter, write notes and examples for each concept under the tabs.. Reading and Writing. Chapter 14. Probability. 753. TM. For more information about Foldables, see Teaching Mathematics with Foldables.. Organization of Data with a Concept Map Begin with the central chapter theme of Probability as the title and have students record key words and phrases under the four tabs labeled outcomes, permutations, combinations, and compound events. Students use their Foldables to take notes, define terms, record concepts, and write examples. Foldable concept maps make great study aids because students view main ideas, recall what they know, and check their responses by looking under the tabs. Chapter 14 Probability 753.

(9) Lesson Notes. 1 Focus 5-Minute Check Transparency 14-1 Use as a quiz or review of Chapter 13. Mathematical Background notes are available for this lesson on p. 752C.. Counting Outcomes • Count outcomes using a tree diagram. • Count outcomes using the Fundamental Counting Principle.. Vocabulary • • • •. tree diagram sample space event Fundamental Counting Principle • factorial. are possible win-loss records counted in football? Ask students: • How many different ways can the team end up with a 3–0 record? one way • Why are there three different ways the team could end up with a 2–1 record? They could win, win, lose; win, lose, win; or lose, win, win. • Sports Only in the last few years have college football teams been able to play overtime to break a tie score. Before the overtime rule was established, there were three possible outcomes for a college football game; win, lose, or tie. How many different records could the team have in three games if ties were possible outcomes? 27. are possible win–loss records counted in football? The championship in the Game 1 Atlantic Coast Conference is decided by the number of conference wins. If there is win a tie in conference wins, then the team with more nonconference wins is champion. If Florida State plays 3 nonconference games, lose the diagram at the right shows the different records they could have for those games.. Game 2 win lose. win lose. win lose win lose. Win–Loss Record 3–0 2–1 2–1 1–2. win lose win lose. 2–1 1–2 1–2 0–3. Game 3. TREE DIAGRAMS One method used for counting the number of possible outcomes is to draw a tree diagram. The last column of a tree diagram shows all of the possible outcomes. The list of all possible outcomes is called the sample space, while any collection of one or more outcomes in the sample space is called an event.. Example 1 Tree Diagram A football team uses red jerseys for road games, white jerseys for home games, and gray jerseys for practice games. The team uses gray or black pants, and black or white shoes. Use a tree diagram to determine the number of possible uniforms. Jersey. Pants Gray. Red Black. Gray White Black. Gray Gray Black. Shoes. Outcomes. Black White Black White. RGB RGW RBB RBW. Black White Black White. WGB WGW WBB WBW. Black White Black White. GGB GGW GBB GBW. The tree diagram shows that there are 12 possible uniforms. 754. Chapter 14 Probability. Resource Manager Workbook and Reproducible Masters Chapter 14 Resource Masters • Study Guide and Intervention, pp. 831–832 • Skills Practice, p. 833 • Practice, p. 834 • Reading to Learn Mathematics, p. 835 • Enrichment, p. 836. Parent and Student Study Guide Workbook, p. 107. Transparencies 5-Minute Check Transparency 14-1 Answer Key Transparencies. Technology Interactive Chalkboard.

(10) THE FUNDAMENTAL COUNTING PRINCIPLE The number of possible uniforms in Example 1 can also be found by multiplying the number of choices for each item. If the team can choose from 3 different colored jerseys, 2 different colored pants, and 2 different colored pairs of shoes, there are 3 2 2 or 12 possible uniforms. This example illustrates the Fundamental Counting Principle.. Fundamental Counting Principle If an event M can occur in m ways and is followed by an event N that can occur in n ways, then the event M followed by event N can occur in m n ways.. The Uptown Deli offers a lunch special in which you can choose a sandwich, a side dish, and a beverage. If there are 10 different sandwiches, 12 different side dishes, and 7 different beverages from which to choose, how many different lunch specials can you order? Multiply to find the number of lunch specials.. 12. . 7. number of specials. . In-Class Example. 840. The number of different lunch specials is 840. wheat. Example 3 Counting Arrangements Mackenzie is setting up a display of the ten most popular video games from the previous week. If she places the games side-by-side on a shelf, in how many different ways can she arrange them? The number of ways to arrange the games can be found by multiplying the number of choices for each position. • Mackenzie has ten games from which to choose for the first position. • After choosing a game for the first position, there are nine games left from which to choose for the second position. • There are now eight choices for the third position. • This process continues until there is only one choice left for the last position.. rye. Let n represent the number of arrangements. There are 3,628,800 different ways to arrange the video games.. Side Item Outcomes chips WBC bologna brownie WBB fruit WBF chips WTC turkey brownie WTB fruit WTF chips WHC ham brownie WHB fruit WHF chips RBC bologna brownie RBB fruit RBF chips RTC turkey brownie RTB fruit RTF chips RHC ham brownie RHB fruit RHF. In-Class Examples. The expression n 10 9 8 7 6 5 4 3 2 1 used in Example 3 can be written as 10! using a factorial.. The expression n!, read n factorial, where n is greater than zero, is the product of all positive integers beginning with n and counting backward to 1.. • Symbols n! n (n 1) (n 2) … 3 2 1 • Example 5! 5 4 3 2 1 or 120 By definition, 0! 1. Lesson 14-1 Counting Outcomes. Power Point®. 2 The Too Cheap computer com-. Factorial. www.algebra1.com/extra_examples. Meat. THE FUNDAMENTAL COUNTING PRINCIPLE. n 10 9 8 7 6 5 4 3 2 1 or 3,628,800. • Words. Power Point®. concession stand sells sandwiches on either wheat or rye bread. The sandwiches come with bologna, turkey, or ham; and either chips, a brownie, or fruit. Use a tree diagram to determine the number of possible sandwich combinations. 18 possible combinations Bread. . beverage choices. . . . 10. side dish choices. TREE DIAGRAMS. 1 At football games, a student. Example 2 Fundamental Counting Principle. sandwich choices. 2 Teach. 755. pany sells custom made personal computers. Customers have a choice of 11 different hard drives, 6 different keyboards, 4 different mice, and 4 different monitors. How many different custom computers can you order? 11 6 4 4 1056 different custom computers.. 3 There are 8 students in the Differentiated Instruction Logical Before you introduce the Fundamental Counting Principle, ask students to study Example 1 and make a conjecture about any relationship between the number of possible jerseys, pants, and shoes and the number of outcomes. Guide students to see that the number of outcomes is the product of the number of choices.. Algebra Club at Central High School. The students want to stand in a line for their yearbook picture. How many different ways could the 8 students stand for their picture? 87654321 40,320 ways they could stand. Lesson 14-1 Counting Outcomes 755.

(11) In-Class Examples. Example 4 Factorial. Power Point®. Find the value of each expression.. 4 Find the value of 9!.. a. 6!. 9! 9 8 7 6 5 4 3 2 1 362,880. 6! 6 5 4 3 2 1 Definition of factorial 720. Teaching Tip. Explain to students that in Part b of Example 5, even though Zach and Kurt only get to ride 8 of the roller coasters, there are still 12 to choose from when they ride the first one, 11 for the second one, 10 for the third one, and so on.. 5 OUTDOORS Jill and Miranda are going to a National Park for their vacation. Near the campground where they are staying, there are 8 hiking trails. a. How many different ways can they hike all the trails if they hike each trail only once? 40,320 b. If they only have time to hike on 5 of the trails, how many ways can they do this? 6720. Simplify.. b. 10! 10! 10 9 8 7 6 5 4 3 2 1 Definition of factorial 3,628,800. Simplify.. Example 5 Use Factorials to Solve a Problem ROLLER COASTERS Zach and Kurt are going to an amusement park. They cannot decide in which order to ride the 12 roller coasters in the park.. Roller Coasters. a. How many different orders can they ride all of the roller coasters if they ride each once?. In 2000, there were 646 roller coasters in the United States. Type. Use a factorial. 12! 12 11 10 9 8 7 6 5 4 3 2 1 479,001,600. Number. Wood. 118. Steel. 445. Inverted. 35. Stand Up. 10. Suspended. 11. Wild Mouse. 27. Definition of factorial Simplify.. There are 479,001,600 ways in which Zach and Kurt can ride all 12 roller coasters. b. If they only have time to ride 8 of the roller coasters, how many ways can they do this? Use the Fundamental Counting Principle to find the sample space. s 12 11 10 9 8 7 6 5 Fundamental Counting Principle. Source: Roller Coaster Database. 19,958,400. Simplify.. There are 19,958,400 ways for Zach and Kurt to ride 8 of the roller coasters.. 3 Practice/Apply. Concept Check 2. See margin.. 1. OPEN ENDED Give an example of an event that has 7 6 or 42 outcomes.. Sample answer: choosing 2 books from 7 books on a shelf. 2. Draw a tree diagram to represent the outcomes of tossing a coin three times. 3. Explain what the notation 5! means. 5! 5 • 4 • 3 • 2 • 1. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 14. • include any other item(s) that they find helpful in mastering the skills in this lesson.. Guided Practice. For Exercises 4–6, suppose the spinner at the right is spun three times.. GUIDED PRACTICE KEY. 4. Draw a tree diagram to show the sample space. See pp.. Exercises. Examples. 4–6 7 8. 1–3 4 5. Application. 756. 5. How many outcomes are possible? 64. 795A–795B.. 6. How many outcomes involve both green and blue? 18 7. Find the value of 8!. 40,320 8. SCHOOL In a science class, each student must choose a lab project from a list of 15, write a paper on one of 6 topics, and give a presentation about one of 8 subjects. How many different ways can students choose to do their assignments? 720. Chapter 14 Probability. Answers 2. Toss 1. Toss 2 H. H T H T T. Toss 3 H T H T H T H T. Outcomes HHH HHT HTH HTT THH THT TTH TTT. 19. 0 4 1. 0 5 1. 756. Chapter 14 Probability. 7 8 9 7 8 9 7 8 9 7 8 9. 407 408 409 417 418 419 507 508 509 517 518 519. 20. Columbus in three games : C-C-C; Columbus in four games: C-C-D-C, C-D-C-C, D-C-C-C; Columbus in five games: C-C-D-D-C, C-D-C-D-C, C-D-D-C-C, D-C-C-D-C, D-C-D-C-C, D-D-C-C-C; D.C. in three games: D-D-D; D.C. in four games: C-D-D-D, D-C-D-D, D-D-C-D; D.C. in five games: C-C-D-D-D, C-D-C-D-D, C-D-D-C-D, D-C-C-D-D, D-C-D-C-D, D-D-C-C-D.

(12) ★ indicates increased difficulty. NAME ______________________________________________ DATE. For Exercises. See Examples. 9, 10, 19 11–14 15–18, 20–22. 1 4 2, 3, 5. p. 831 (shown) Counting Outcomes and p. 832. Tree Diagrams One method used for counting the number of possible outcomes of an event is to draw a tree diagram. The last column of the tree diagram shows all of the possible outcomes. The list of all possible outcomes is called the sample space, and a specific outcome is called an event.. Draw a tree diagram to show the sample space for each event. Determine the number of possible outcomes. 9–10. See pp. 795A–795B for diagrams. 9. earning an A, B, or C in English, Math, and Science classes 9 10. buying a computer with a choice of a CD-ROM, a CD recorder, or a DVD drive, one of 2 monitors, and either a printer or a scanner 12. Example 1 Suppose you can set up a stereo system with a choice of video, DVD, or laser disk players, a choice of cassette or graphic equalizer audio components, and a choice of single or dual speakers. Draw a tree diagram to show the sample space. Player. Audio. Extra Practice See page 851.. Speaker Single Dual Single Dual Single Dual Single Dual Single Dual Single Dual. cassette video. Find the value of each expression. 11. 4! 24. 12. 7! 5040. graphic equalizer cassette. DVD. 13. 11! 39,916,800 14. 13!. graphic equalizer cassette. laser disk. 6,227,020,800 15. Three dice, one red, one white, and one blue are rolled. How many outcomes are possible? 216. graphic equalizer. Outcomes VCS VCD VGS VGD DCS DCD DGS DGD LCS LCD LGS LGD. Example 2 A food stand offers ice cream cones in vanilla or chocolate flavors. It also offers fudge or caramel toppings, and it uses sugar or cake cones. Use a tree diagram to determine the number of possible ice cream cones. Flavor. Toppings fudge. vanilla caramel fudge chocolate caramel. The tree diagram shows that there are 12 ways to set up the stereo system.. Cone sugar cake sugar cake sugar cake sugar cake. Outcomes VFS VFC VCS VCC CFS CFC CCS CCC. Lesson 14-1. Homework Help. ____________ PERIOD _____. Study Guide andIntervention Intervention, 14-1 Study Guide and. Practice and Apply. The tree diagram shows that there are 8 possible ice cream cones.. Exercises The spinner at the right is spun twice. 1. Draw a tree diagram to show the sample space. A A. 16. How many outfits are possible if you choose one each of 5 shirts, 3 pairs of pants, 3 pairs of shoes, and 4 jackets? 180. B. B C. D. A. B. C C. D. A. B. A. B. D. C. D C. D. A. B. C. D. AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD. 2. How many outcomes are possible? 16. 17. TRAVEL Suppose four different airlines fly from Seattle to Denver. Those same four airlines and two others fly from Denver to St. Louis. If there are no direct flights from Seattle to St. Louis, in how many ways can a traveler book a flight from Seattle to St. Louis? 24. A pizza can be ordered with a choice of sausage, pepperoni, or mushrooms for toppings, a choice of thin or pan for the crust, and a choice of medium or large for the size. Toppings Crust Size Outcomes sausage pan. 4. How many pizzas are possible? 12. thin pepperoni pan thin mushrooms pan. NAME ______________________________________________ DATE. ★ COMMUNICATIONS For Exercises 18 and 19, use the following information.. Skills Practice, 14-1 Practice (Average). A new 3-digit area code is needed in a certain area to accommodate new telephone numbers. 18. If the first digit must be odd, the second digit must be a 0 or a 1, and the third digit can be anything, how many area codes are possible? 100. ____________ PERIOD _____. Draw a tree diagram to show the sample space for each event. Determine the number of possible outcomes. 1. dining at an Italian, Mexican, or French restaurant, for lunch, early bird (early dinner special), or dinner, and with or without dessert 18 Style. Time. Dessert dessert no dessert dessert no dessert dessert no dessert dessert no dessert dessert no dessert dessert no dessert dessert no dessert dessert no dessert dessert no dessert. lunch early bird dinner lunch Mexican. ★ SOCCER For Exercises 20–22, use the following information.. early bird dinner lunch. The Columbus Crew are playing the D.C. United in a best three-out-of-five championship soccer series.. STM STL SPM SPL PTM PTL PPM PPL MTM MTL MPM MPL. p. 833 and Practice, p. 834 (shown) Counting Outcomes. Italian. 19. Draw a tree diagram to show the different area codes using 4 or 5 for the first digit, 0 or 1 for the second digit, and 7, 8, or 9 for the third digit. See margin.. medium large medium large medium large medium large medium large medium large. thin. 3. Draw a tree diagram to show the sample space.. French. early bird dinner. Outcomes ILD ILN IED IEN IDD IDN MLD MLN MED MEN MDD MDN FLD FLN FED FEN FDD FDN. There are 18 possible outcomes.. 20. What are the possible outcomes of the series? See margin.. Find the value of each expression.. 21. How many outcomes require only four games be played to determine the champion? 6. 2. 5! 120. 3. 8! 40,320. 4. 10! 3,628,800. 5. 12! 479,001,600. 6. How many different vacation plans are possible when choosing one each of 12 destinations, 3 lengths of stay, 5 travel options, and 4 types of accommodations? 720 7. How many different ways can you arrange your work if you can choose from 7 weekly schedules, 6 daily schedules, and one of 3 types of duties? 126. 22. How many ways can D.C. United win the championship? 10. 8. How many different ways can you treat a minor cut if you can choose from 3 methods of cleansing the cut, 5 antibiotic creams, 2 antibacterial sprays, and 6 types of bandages? 180 9. TESTING A teacher gives a quick quiz that has 4 true/false questions and 2 multiple choice questions, each of which has 5 answer choices. In how many ways can the quiz be answered if one answer is given for each question? 400. 23. CRITICAL THINKING To get to and from school, Tucker can walk, ride his bike, or get a ride with a friend. Suppose that one week he walked 60% of the time, rode his bike 20% of the time, and rode with his friend 20% of the time. How many outcomes represent this situation? Assume that he returns home the same way that he went to school. 20. CLASS RINGS Students at Pacific High can choose class rings in one each of 8 styles, 5 metals, 2 finishes, 14 stones, 7 cuts of stone, 4 tops, 3 printing styles, and 30 inscriptions.. 24. WRITING IN MATH. Reading 14-1 Readingto to Learn Learn Mathematics. 10. How many different choices are there for a class ring? 2,822,400 11. If a student narrows the choice to 2 styles, 3 metals, 4 cuts of stone, and 5 inscriptions (and has already made the remaining decisions), how many different choices for a ring remain? 120 NAME ______________________________________________ DATE. Answer the question that was posed at the beginning of the lesson. See margin.. Mathematics, p. 835 Counting Outcomes. Pre-Activity. Include the following in your answer: • a few sentences describing how a tree diagram can be used to count the wins and losses of a football team, and • a demonstration of how to find the number of possible outcomes for a team that plays 4 home games. Lesson 14-1 Counting Outcomes. 757. ELL. How are possible win/loss football records counted? Read the introduction to Lesson 14-1 at the top of page 754 in your textbook. Then complete the diagram. Game 2. Game 1. win win. Game 3 win lose win lose. lose. win lose win. win lose lose. lose. Outcomes win-win-win. win-win-lose win-lose-win win-lose-lose lose-win-win lose-win-lose lose-lose-win lose-lose-lose. Lesson 14-1. How are possible win–loss records counted in football?. www.algebra1.com/self_check_quiz. ____________ PERIOD _____. Reading the Lesson Use the tree diagram above for Exercises 1–4. 1. What is the sample space?. win-win-win, win-win-lose, win-lose-win, win-lose-lose, lose-win-win, lose-win-lose, lose-lose-win, lose-lose-lose. 24. Sample answer: You can make a chart showing all possible outcomes to help determine a football team’s record. Answers should include the following. • a tree diagram or calculations to show 16 possible outcomes. NAME ______________________________________________ DATE. 14-1 Enrichment Enrichment,. ____________ PERIOD _____. p. 836. 2. Name two different outcomes.. Sample answer: win-win-lose, win-lose-win 3. Three different outcomes result in a win/loss record of 2-1. What are they?. Pascal’s Triangle. win-win-lose, win-lose-win, lose-win-win. Pascal's Triangle is a pattern of numbers used at many levels of mathematics. It is named for Blaise Pascal, a seventeenth-century French mathematician who discovered several applications of the pattern. However, records of the triangle have been traced as far back as twelfth-century China and Persia. In the year –-Shìjié 1303, the Chinese mathematician Zhu wrote The Precious Mirror of the Four Elements, in which he described how the triangle could be used to solve polynomial equations. The figure at the right is adapted from the original Chinese manuscript. In the figure, some circles are empty while others contain Chinese symbols.. At the right, a portion of Pascal’s Triangle is shown using Hindu-Arabic numerals.. 4. Use the Fundamental Counting Principle to complete the chart. Game 1 Number of Choices. 2. Game 2 . 2. Game 3 . 2. Number of Outcomes . 8. Helping You Remember 5. Suppose you are training the new disc jockey for a school radio station. He has chosen 10 selections to play from a new CD. How could you use factorials to explain to him the number of different ways the selections could be played?. Multiply the number of possible choices for each slot on the playlist. There are 10 choices for the first song, nine choices for the second song, and so on. So the selections can be played in 10 9 8 7 6 5 4 3 2 1 3,628,800 ways.. 1 1. 1. Lesson 14-1 Counting Outcomes 757.

(13) Standardized Test Practice. About the Exercises… Organization by Objective • Tree Diagrams: 9–10, 19 • The Fundamental Counting Principle: 15–17, 20–22 Odd/Even Assignments Exercises 9–16 are structured so that students practice the same concepts whether they are assigned odd or even problems.. 25. Evaluate 9!. A A. B. 362,880. C. 40,320. A. B. 96. C. 144. Mixed Review. A B 30. 40. 50. 60. 70. 80. 90. 27. Determine the least value, greatest value, lower quartile, upper quartile, and median for each plot. A: 32, 88, 44, 85, 60; B: 38, 86, 48, 74, 64 28. Which set of data contains the least value? A 29. Which plot has the smaller interquartile range? B 30. Which plot has the greater range? A Stem. (Lesson 13-4). 4 Assess. 31. Find the range of the data. 79 32. What is the median? 57.5 33. Determine the upper quartile, lower quartile, and interquartile range of the data. 73.5; 39.5; 34.0. Open-Ended Assessment. will learn about permutations and combinations in Lesson 14-2 and use them to determine the probability of a given event. Use Exercises 46–51 to determine your students’ familiarity with simple probability.. 384. For Exercises 27–30, use box-and-whisker plots A and B. (Lesson 13-5). For Exercises 31–34, use the stem-and-leaf plot.. PREREQUISITE SKILL Students. D. 288. Maintain Your Skills. Basic: 9–17 odd, 23–51 Average: 9–17 odd, 18, 19, 23–51 Advanced: 10–16 even, 20–45 (optional: 46–51). Getting Ready for Lesson 14-2. 8. 26. A car manufacturer offers a sports car in 4 different models with 6 different option packages. Each model is available in 12 different colors. How many different possibilities are available for this car? C. Assignment Guide. Modeling Have students use construction paper to model sandwich ingedients, such as different types of bread, meat, and vegetables. Have students first calculate how many different sandwiches they could make. Then have them make the sandwiches to confirm their calculations.. D. 36. 34. Identify any outliers. none. Study Tip Deck of Cards In this text, a standard deck of cards always means a deck of 52 playing cards. There are 4 suits—clubs (black), diamonds (red), hearts (red), and spades (black)—with 13 cards in each suit.. Find each sum or difference. (Lesson 12-7) 2x 1 x 4 5x2 8x 6 35. 36. 3x 1 x 2 (3x 1)(x 2) z 2 3z 1 3z 2 37. 38. z2 4 3z 6 3z 6. 758. Chapter 14 Probability. 9 30 30. m2 2mn n2 1 1 mn m2 n2 mn m2 n2. 2 39. 52n 28 20 . 22 . 40. 5x2 7 2x. 7. 41. x2x4 7. Solve each equation by completing the square. Round to the nearest tenth if necessary. (Lesson 10-3) 43. n2 8n 5 0 8.6, 0.6. 42. b2 6b 4 0 0.8, 5.2 x2. 11x 17 0 1.4, 12.4. 45. 2p2 10p 3 0 4.7, 0.3. PREREQUISITE SKILL One card is drawn at random from a standard deck of cards. Find each probability. (To review simple probability, see Lesson 2-6.) 1 1 1 46. P(10) 47. P(ace) 48. P(red 5) 13 13 26 2 1 5 49. P(queen of clubs) 50. P(even number) 51. P(3 or king) 52 13 13. Answers (page 759). 758. 1 4 5 4 8 9 8 6 1. 2n 3 4n 3 2n 6 n3 n3. Chapter 14 Probability. 1. yes; sample answer: Front St., Main St., Second Ave., State St., Elm St., First Ave., Town St. 4. Yes; all nodes can be connected without retracing an edge. 5. No; all nodes cannot be connected without retracing an edge. 6. Yes; all nodes can be connected without retracing an edge.. 0 4 6 6 1 0. Solve each equation. (Lesson 11-3). 44.. Getting Ready for the Next Lesson. 3 4 5 6 7 8 9 10. Leaf. 7a.. 8. Sample answer: If you follow the edges of a graph, you should cover each edge only once..

(14) Algebra Activity. A Follow-Up of Lesson 14-1. A Follow-Up of Lesson 14-1. Finite Graphs. Getting Started. The City Bus Company provides daily bus service between City College and Southland Mall, City College and downtown, downtown and Southland Mall, downtown and City Park, and City Park and the zoo. The daily routes can be represented using a finite graph like the one at the right. The graph is called a network, and each point on the graph is called a node. The paths connecting the nodes are called edges . A network is said to be traceable if all of the nodes can be connected, and each edge can be covered exactly once when the graph is used.. Zoo. edge. Downtown. node. City Park. City College. The graph represents the streets on Alek’s newspaper route. To get his route completed as quickly as possible, Alek would like to ride his bike down each street only once. • Copy the graph onto your paper. • Beginning at Alek’s home, trace over his route without lifting your pencil. Remember to trace each edge only once. • Compare your graph with those of your classmates.. Teach. Southland Mall. Alek’s Home. Collect the Data. Town St. First Ave.. State St. Elm St.. Second Ave.. Front St.. Main St.. Analyze the Data 1. Is Alek’s route traceable? If so, describe his route. See margin. 2. Is there more than one traceable route that begins at Alek’s house? If so,. how many? yes; 4 3. Suppose it does not matter where Alek starts his route. How many traceable. routes are possible now? 8 Determine whether each graph is traceable. Explain your reasoning. 4–6. See margin. 4. 5. 6.. 7. The campus for Centerburgh High School has five buildings built around the. edge of a circular courtyard. There is a sidewalk between each pair of buildings. a. Draw a graph of the campus. See margin. b. Is the graph traceable? yes c. Suppose that there is not a sidewalk between the pairs of adjacent buildings. Is it possible to reach all five buildings without walking down any sidewalk more than once? yes 8. Make a conjecture for a rule to determine whether a graph is traceable. See margin. Investigating Slope-Intercept Form 759 Algebra Activity Finite Graphs 759. Resource Manager Teaching Algebra with Manipulatives. • p. 206 (student recording sheet). Objective Determine whether finite graphs are traceable. Materials paper pencil. • If blank transparencies are available, have students use the transparencies to trace the graphs. • It is important for students to compare their tracings of Alek’s route with those of their classmates to see that there is more than one possible route. • When students answer question 3, they will find that there are only two possible starting points on this graph; Alek’s house and the other end of State Street. • Help students make the connection between finding possible routes on traceable graphs and counting outcomes. When students find that there are two possible starting points, each with four possible routes, they can use the Fundamental Counting Principle to determine that there are eight total routes.. Assess Work with students to formulate a conjecture about when a graph is traceable and when it is not.. Study Notebook You may wish to have students summarize this activity and what they learned from it.. Algebra Activity Finite Graphs 759.

(15) Permutations and Combinations. Lesson Notes. 1 Focus 5-Minute Check Transparency 14-2 Use as a quiz or review of Lesson 14-1.. • Determine probabilities using permutations. • Determine probabilities using combinations.. Vocabulary • permutation • combination. Mathematical Background notes are available for this lesson on p. 752C. can combinations be used to form committees? Ask students: • Suppose Senators Kennedy, Jeffords, and Collins are three of the members of the committee. Explain why the order in which they are selected does not matter. No matter how the three senators are selected, all three are still on the committee. • If there were more Democrats in the Senate than Republicans, would the number of ways the committee members could be selected be affected? Explain. No. There will still be 18 committee members, which does not change the ways in which they could be selected.. can combinations be used to form committees? The United States Senate forms various committees by selecting senators from both political parties. The Senate Health, Education, Labor, and Pensions Committee of the 106th Congress was made up of 10 Republican senators and 8 Democratic senators. How many different ways could the committee have been selected? The members of the committee were selected in no particular order. This is an example of an arrangement called a combination.. Senate Health, Education, Labor, and Pensions Committee 46. Democrats. 54. Republicans. PERMUTATIONS An arrangement or listing in which order or placement is important is called a permutation .. Example 1 Tree Diagram Permutation EMPLOYMENT The manager of a coffee shop needs to hire two employees, one to work at the counter and one to work at the drive-through window. Katie, Bob, Alicia, and Jeremiah all applied for a job. How many possible ways are there for the manager to place the applicants?. Study Tip Common Misconception When arranging two objects A and B using a permutation, the arrangement AB is different from the arrangement BA.. Use a tree diagram to show the possible arrangements. Counter Katie (K). Bob (B). Alicia (A). Jeremiah (J). Drive-Through. Outcomes. Bob. KB. Alicia. KA. Jeremiah. KJ. Katie. BK. Alicia. BA. Jeremiah. BJ. Jeremiah. AJ. Katie. AK. Bob. AB. Katie. JK. Bob. JB. Alicia. JA. There are 12 different ways for the 4 applicants to hold the 2 positions. 760. Chapter 14 Probability. Resource Manager Workbook and Reproducible Masters Chapter 14 Resource Masters • Study Guide and Intervention, pp. 837–838 • Skills Practice, p. 839 • Practice, p. 840 • Reading to Learn Mathematics, p. 841 • Enrichment, p. 842 • Assessment, p. 875. Parent and Student Study Guide Workbook, p. 108 School-to-Career Masters, p. 27. Transparencies 5-Minute Check Transparency 14-2 Answer Key Transparencies. Technology Interactive Chalkboard.

(16) In Example 1, the positions are in a specific order, so each arrangement is unique. The symbol 4P2 denotes the number of permutations when arranging 4 applicants in 2 positions. You can also use the Fundamental Counting Principle to determine the number of permutations.. 4P2. . 4 21 21. 4 3 4321 21 4! 2!. . PERMUTATIONS. ways to choose second employee. . ways to choose first employee. 2 Teach. . In-Class Examples. 3. 21 1 21. Reading Tip. Permutations of n objects taken r at a time can also be written as P(n, r).. Multiply.. 1 Ms. Baraza asks pairs of stu-. 4 3 2 1 4!, 2 1 2!. In general, nPr is used to denote the number of permutations of n objects taken r at a time.. Permutation • Words. The number of permutations of n objects taken r at a time is the quotient of n! and (n r)!.. • Symbols. nPr. n! (n r)!. Example 2 Permutation Find 10P6. n! (n r)! 10! 10P6 (10 6)! 10! 10P6 4 ! nPr. . Definition of nPr. Subtract.. There are 1680 permutations of 8 objects taken 4 at a time.. Definition of factorial. Teaching Tip. 1. 4321. Point out to students that the numbers 0, 3, and 8 are not possible choices.. 1. 10 9 8 7 6 5 or 151,200 Simplify.. There are 151,200 permutations of 10 objects taken 6 at a time.. 3 Shaquille has a 5-digit pass code to access his e-mail account. The code is made up of the even digits, 2, 4, 6, 8, and 0. Each digit can be used only once.. Permutations are often used to find the probability of events occurring.. Example 3 Permutation and Probability A word processing program requires a user to enter a 7-digit registration code made up of the digits 1, 2, 4, 5, 6, 7, and 9. Each number has to be used, and no number can be used more than once.. Study Tip Permutations The number of permutations of n objects taken n at a time is n!. n Pn n!. a. How many different pass codes could Shaquille have? 120. a. How many different registration codes are possible? Since the order of the numbers in the code is important, this situation is a permutation of 7 digits taken 7 at a time. n! (n r)! 7! 7P7 (7 7)! nPr. dents to go in front of her Spanish class to read statements in Spanish, and then to translate the statement into English. One student is the Spanish speaker and one is the English speaker. If Ms. Baraza has to choose between Jeff, Kathy, Guillermo, Ana, and Patrice, how many different ways can Ms. Baraza pair the students? 20 ways. 2 Find 8P4.. n 10, r 6. 10 9 8 7 6 5 4 3 2 1 10P6 10P6. Power Point®. . b. What is the probability that the first two digits of his code are both greater than 5?. Definition of permutation n 7, r 7. 7654321 or 5040 7P7 1. 1 or 10% 10. Definition of factorial. There are 5040 possible codes with the digits 1, 2, 4, 5, 6, 7, and 9.. www.algebra1.com/extra_examples. Lesson 14-2 Permutations and Combinations. 761. Differentiated Instruction Visual/Spatial Have students write the digits 1, 2, 4, 5, 6, 7, and 9 on index cards. Have them rearrange the cards in different ways to help them visualize how the permutation formula relates to the cards. Compare their results with the calculations in Example 3.. Lesson 14-2 Permutations and Combinations 761.

(17) COMBINATIONS. In-Class Example. b. What is the probability that the first three digits of the code are even numbers? Power Point®. Reading Tip. Combinations of n objects taken r at a time can also be written C(n, r).. Study Tip. Use the Fundamental Counting Principle to determine the number of ways for the first three digits to be even.. Look Back. • There are 3 even digits and 4 odd digits.. To review probability, see Lesson 2-6.. • The number of choices for the first three digits, if they are even, is 3 2 1. • The number of choices for the remaining odd digits is 4 3 2 1.. 4 TEST ITEM Customers at. • The number of favorable outcomes is 3 2 1 4 3 2 1 or 144. There are 144 ways for this event to occur out of the 5040 possible permutations.. Tony’s Pizzeria can choose 4 out of 12 toppings for each pizza for no extra charge. How many different combinations of pizza toppings can be chosen? A. 144 5040 1 35. P(first 3 digits even) . ← number of favorable outcomes ← number of possible outcomes. Simplify.. 1 35. The probability that the first three digits of the code are even is or about 3%.. A 495 B 792 C 11,880 D 95,040. COMBINATIONS An arrangement or listing in which order is not important is called a combination. For example, if you are choosing 2 salad ingredients from a list of 10, the order in which you choose the ingredients does not matter.. Combination. Teaching Tip. After students learn how to calculate combinations and have worked Example 4, discuss the difference between combinations and permutations. In Example 4, the order in which the students are chosen does not matter because the positions for which they are being chosen are the same. They are all four going to be members of the student council, with the same duties. However, if the homeroom was choosing 4 out of 7 students to be president, vice president, secretary, and treasurer of the student council, then the order in which they are chosen does matter.. • Words. The number of combinations of n objects taken r at a time is the quotient of n! and (n r)!r!.. • Symbols. nCr. n! (n r)!r!. . Standardized Example 4 Combination Test Practice Multiple-Choice Test Item The students of Mr. DeLuca’s homeroom had to choose 4 out of the 7 people who were nominated to serve on the Student Council. How many different groups of students could be selected? A. 840. B. 210. C. 35. D. 24. Read the Test Item The order in which the students are chosen does not matter, so this situation represents a combination of 7 people taken 4 at a time.. Test-Taking Tip Read each question carefully to determine whether the situation involves a permutation or a combination. Often, the answer choices include examples of both.. Solve the Test Item n! (n r)!r!. nCr. . 7C4. . 7! (7 4)!4!. Definition of combination n 7, r 4 1. 7654321 3214321. 765 321. or 35. Definition of factorial. 1. Simplify.. There are 35 different groups of students that could be selected. Choice C is correct. 762. Chapter 14 Probability. Example 4 Tell students to pay close attention to the order in which the information is given in word problems. Because 4 is listed before 7 in the problem, students might mistakenly think that n 4 and r 7. Choice A is the correct number of permutations for 7 P4. However this problem involves combinations.. Standardized Test Practice. 762. Chapter 14 Probability.

(18) Combinations and the products of combinations can be used to determine probabilities.. In-Class Example. Power Point®. Teaching Tip. Multi-part problems such as Example 5 often contain more information than is necessary to solve the first part of the problem. For example, to solve part a, it is not necessary to know that the group of students includes 7 seniors, 5 juniors, and 4 sophomores. The only necessary information is that 12 students are to be chosen out of a group of 16.. Example 5 Use Combinations SCHOOL A science teacher at Sunnydale High School needs to choose 12 students out of 16 to serve as peer tutors. A group of 7 seniors, 5 juniors, and 4 sophomores have volunteered to be tutors. a. How many different ways can the teacher choose 12 students? The order in which the students are chosen does not matter, so we must find the number of combinations of 16 students taken 12 at a time. n! (n r)!r! 16! 16C12 (16 12)!12! 16! 4!12! 1 16 15 14 13 12! 4! 12! nCr. . Definition of combination n 16, r 12. 5 MONEY Diane has a bag full. 16 12 4. of coins. There are 10 pennies, 6 nickels, 4 dimes, and 2 quarters in the bag.. Divide by the GCF, 12!.. 1. 43,680 24. or 1820. Simplify.. a. How many different ways can Diane pull four coins out of the bag? There are 7315 ways to pull 4 coins out of a bag of 22.. There are 1820 ways to choose 12 students out of 16. b. If the students are chosen randomly, what is the probability that 4 seniors, 4 juniors, and 4 sophomores will be selected?. b. What is the probability that she will pull two pennies and two nickels out of the bag? The probability that Diane will select two pennies and two. There are three questions to consider. • How many ways can 4 seniors be chosen from 7? • How many ways can 4 juniors be chosen from 5? • How many ways can 4 sophomores be chosen from 4?. 675 7313. Using the Fundamental Counting Principle, the answer can be determined with the product of the three combinations. ways to choose 4 juniors out of 5. ways to choose 4 sophomores out of 4. . ways to choose 4 seniors out of 7. nickels is , or about 9%.. (7C4). Study Tip. . (5C4). . 7! 5! (7 4)!4! (5 4)!4! 7! 5! 4! 3!4! 1!4! 0!4! 765 5 3! 1. 4! (4 4)!4!. (7C4)(5C4)(4C4) . Combinations The number of combinations of n objects taken n at a time is 1. nCn 1. Concept Check. (4C4) Definition of combination Simplify. Divide by the GCF, 4!.. 175. Simplify.. There are 175 ways to choose this particular combination out of 1820 possible combinations. 175 1820 5 52. P(4 seniors, 4 juniors, 4 sophomores) . ← number of favorable outcomes ← number of possible outcomes. Simplify.. The probability that the science teacher will randomly select 4 seniors, 4 juniors,. Combinations Explain why you would expect the number of combinations of n items taken r at a time to be less than the number of permutations of n items taken r at a time. Sample answer: The number of combinations is less because order does not matter. For example, in a permutation, AB is different from BA because order matters. In a combination, AB and BA are the same.. 5 52. and 4 sophomores is or about 10%.. www.algebra1.com/extra_examples. Lesson 14-2 Permutations and Combinations. 763. The details in Example 5 might be confusing for some students. Consider representing the problem information on an overhead transparency. Assign three different colors to represent seniors, juniors, and sophomores. Then draw groups of colored dots to represent the 16 students. In part b of Example 5, show students how to focus on one group of students at a time by examining only one color of dot.. New. Lesson 14-2 Permutations and Combinations 763.

(19) 3 Practice/Apply. Concept Check 1–3. See margin.. 2. Demonstrate and explain why nCr 1 whenever n r. What does nPr always equal when n r?. Study Notebook Have students— • add the definitions/examples of the vocabulary terms to their Vocabulary Builder worksheets for Chapter 14. • include an example of how to find the number of permutations and number of combinations of n items taken r at a time. • include any other item(s) that they find helpful in mastering the skills in this lesson.. 1. OPEN ENDED Describe the difference between a permutation and a combination. Then give an example of each.. 3. FIND THE ERROR Eric and Alisa are taking a trip to Washington, D.C. Their tour bus stops at the Lincoln Memorial, the Jefferson Memorial, the Washington Monument, the White House, the Capitol Building, the Supreme Court, and the Pentagon. Both are finding the number of ways they can choose to visit 5 of these 7 sites.. Alisa. Eric 7! 7C5 = 2! or 2520. 7C5. 7! = or 21 2! 5!. Who is correct? Explain your reasoning.. Guided Practice GUIDED PRACTICE KEY Exercises. Examples. 4, 5, 10, 13 6–9, 11, 12. 1, 4 2, 3, 5. Determine whether each situation involves a permutation or combination. Explain your reasoning. 4. Combination; order is not important. 4. choosing 6 books from a selection of 12 for summer reading 5. choosing digits for a personal identification number. Permutation; order is important. Evaluate each expression. 6. 8P5 6720. 7. 7C5 21. 9. (6C2)(4C3) 60. 8. (10P5)(3P2). 181,440. FIND THE ERROR First, students must determine whether Eric and Alisa need to find the number of permutations or combinations. Since the order in which the bus visits the sites does not matter, they need to find the number of combinations. Next, students must examine whether Eric or Alisa used the correct procedure to find the number of combinations.. Answers. For Exercises 10–12, use the following information. The digits 0 through 9 are written on index cards. Three of the cards are randomly selected to form a 3-digit code.. 10. Permutation; the order of the digits is important.. Standardized Test Practice. since 0! 1. n! 0!. or n!,. since 0! 1 3. Alisa; both are correct in that the situation is a combination, but Alisa’s method correctly computes the combination. Eric’s calculations find the number of permutations. 764. 13. A diner offers a choice of two side items from the list with each entrée. How many ways can two items be selected? B C. 15 30. B D. 28 56. Side Items. French fries baked potato cole slaw small salad. mixed vegetables rice pilaf baked beans applesauce. Practice and Apply. n! n! 2. nCn or 1, (n n)!n! 0!n! n! (n n)!. 11. How many different codes are possible? 720 1 12. What is the probability that all 3 digits will be odd? 12. A. 1. Sample answer: Order is important in a permutation but not in a combination. Permutation: the finishing order of a race Combination: toppings on a pizza. nPn. 10. Does this situation represent a permutation or a combination? Explain.. Chapter 14 Probability. Determine whether each situation involves a permutation or combination. Explain your reasoning. 14. team captains for the soccer team Combination; order is not important. 15. three mannequins in a display window Permutation; order is important. 16. a hand of 10 cards from a selection of 52 Combination; order is not important. 17. the batting order of the New York Yankees Permutation; order is important. 764. Chapter 14 Probability.

(20) Homework Help For Exercises. See Examples. 14–21, 34 36, 40 22–33, 35, 37–39, 41–49. 1, 4 2, 3, 5. Extra Practice See page 851.. 18. Permutation; order is important. 20. Combination; order is not important.. 18. first place and runner-up winners for the table tennis tournament 19. a selection of 5 DVDs from a group of eight Combination; order is not important. 20. selection of 2 candy bars from six equally-sized bars. Organization by Objective • Permutations: 14–21, 22–23, 28–31 • Combinations: 14–21, 24–27, 32. 21. the selection of 2 trombones, 3 clarinets, and 2 trumpets for a jazz combo. Combination; order is not important. Evaluate each expression. 22.. 12P3. 1320. 25. 7C3 35. 26.. 15C3 455. 524,160 31. (20P2)(16P4) 16,598,400 32. (3C2)(7C4) 105 28.. 15P3. 2730. 23. 4P1 4 29.. 16P5. 24. 6C6 1. 125,970 30. (7P7)(7P1) 35,280 33. (8C5)(5P5) 6720 27.. 20C8. Odd/Even Assignments Exercises 14–33 are structured so that students practice the same concepts whether they are assigned odd or even problems.. SOFTBALL For Exercises 34 and 35, use the following information. The manager of a softball team needs to prepare a batting lineup using her nine starting players. 34. Is this situation a permutation or a combination? permutation 35. How many different lineups can she make? 362,880. Assignment Guide. SCHOOL For Exercises 36–39, use the following information. Mrs. Moyer’s class has to choose 4 out of 12 people to serve on an activity committee.. Basic: 15–33 odd, 34–39, 50–51, 55–77 Average: 15–33 odd, 36–44, 50, 51, 55–77 Advanced: 14–32 even, 43–71 (optional: 72–77) All: Practice Quiz 1 (1–5). 36. Does the selection of the students involve a permutation or a combination? Explain. Combination; order does not matter. 37. How many different groups of students could be selected? 495. Softball The game of softball was developed in 1888 as an indoor sport for practicing baseball during the winter months. Source: www.encyclopedia.com. About the Exercises…. 38. Suppose the students are selected for the positions of chairperson, activities planner, activity leader, and treasurer. How many different groups of students could be selected? 11,880 39. What is the probability that any one of the students is chosen to be the chairperson? 1 2970 GAMES For Exercises 40–42, use the following information. In your turn of a certain game, you roll five dice at the same time. 40. Do the outcomes of rolling the five dice represent a permutation or a combination? Explain. Combination; order does not matter. 41. How many outcomes are possible? 7776. 1 42. What is the probability that all five dice show the same number on one roll? 1296. BUSINESS For Exercises 43 and 44, use the following information. There are six positions available in the research department of a software company. Of the applicants, 15 are men and 10 are women. 43. In how many ways could 4 men and 2 women be chosen if each were equally qualified? 61,425 44. What is the probability that five women would be selected if the positions were randomly filled? 27 1265 TRACK For Exercises 45 and 46, use the following information. Central High School is competing against West High School at a track meet. Each team entered 4 girls to run the 1600-meter event. The top three finishers are awarded medals. 45. How many different ways can the runners place first, second, and third? 336 46. If all eight runners have an equal chance of placing, what is the probability that the first and second place finishers are from West and the third place finisher is from Central? 1 14% 7 Lesson 14-2 Permutations and Combinations 765 www.algebra1.com/self_check_quiz. Lesson 14-2 Permutations and Combinations 765.

(21) NAME ______________________________________________ DATE. ____________ PERIOD _____. DINING For Exercises 47–49, use the following information. For lunch in the school cafeteria, you can select one item from each category to get the daily combo.. Study Guide andIntervention Intervention, 14-2 Study Guide and p. 837 (shown) and p. 838 Permutations and Combinations. Permutations. An arrangement or listing in which order or placement is important is called a permutation. For example the arrangement AB of choices A and B is different from the arrangement BA of these same two choices. n Pr. Permutations. 6 P2. Entree Burger Sandwich Taco Pizza. Find 6 P2.. n! (n r)!. Definition of n Pr. 6! (6 2)!. n 6, r 2. 6! 4!. Simplify.. . 654321 4321. Definition of factorial. 6 5 or 30. Simplify.. . There are 30 permutations of 6 objects taken 2 at a time.. Example 2 A specific program requires the user to enter a 5-digit password. The digits cannot repeat and can be any five of the digits 1, 2, 3, 4, 7, 8, and 9. a. How many different passwords are possible? . 7 P5. . 48. If a side dish is chosen at random, what is the probability that a student will choose soup? 1 or about 33% 3 49. What is the probability that a student will randomly choose a sandwich and soup? 1 or about 8% 12 CRITICAL THINKING For Exercises 50 and 51, use the following information. Larisa is trying to solve a word puzzle. She needs to arrange the letters H, P, S, T, A, E, and O into a two-word arrangement.. number of favorable outcomes P(first two digits odd) number of possible outcomes. 7! (7 5)!. Since there are 4 odd digits, the number of choices for the first digit is 4, and the number of choices for the second digit is 3. Then there are 5 choices left for the third digit, 4 for the fourth, and 3 for the fifth, so the number of favorable outcomes is 4 3 5 4 3, or 720.. 7654321 21. 7 6 5 4 3 or 2520 There are 2520 ways to create a password.. 720 2520. The probability is 28.6%.. Exercises Evaluate of each expression. 1. 7 P4 840. 2. 12 P7 3,991,680. 3. ( 9 P9 )(16 P2) 87,091,200. 4. A club with ten members wants to choose a president, vice-president, secretary, and treasurer. Six of the members are women, and four are men.. 50. How many different arrangements of the letters can she make? 30,240. a. How many different sets of officers are possible? 5040 b. What is the probability that all officers will be women. 7.1% NAME ______________________________________________ DATE. Skills Practice, 14-2 Practice (Average). ____________ PERIOD _____. p. 839 and Practice, p. and 840 (shown) Permutations Combinations. Determine whether each situation involves a permutation or combination. Explain your reasoning. Sample explanations are given. 1. choosing two dogs from a litter of two males and three females Combination; The. order is not important in the choice of the two dogs. 2. a simple melody formed by playing the notes on 8 different piano keys Permutation;. The sound of the melody depends on the order in which the notes are played. 3. a selection of nine muffins from a shelf of twenty-three Combination; The order. does not matter in the selection of the muffins. 4. the selection of a four-letter acronym (word formed from the initial letters of other words) in which two of the letters cannot be C or P Permutation; The letters of the. acronym must be arranged in a certain order. 5. choosing an alphanumeric password to access a website Permutation; The choice. You can use permutations and combinations to analyze data on U.S. schools. Visit www.algebra1.com/ webquest to continue work on your WebQuest project.. of letters and numbers must be in an exact order for the password to work.. 7. 6 P3 120. 9. 10C9 10. 8. 15 P3 2730. 10. 12C9 220. 11. 7C3 35. 12. 7C4 35. 13. 12C4 495. 14. 13 P3 1716. 15. (8C4)(8C5) 3920. 16. (17 C2)(8C6) 3808. 17. (16C15)(16C1) 256. 18. (8 P3)(8 P2) 18,816. 19. (5 P4)(6 P5) 86,400. 20. (13 P1)(15 P1) 195. 21. (10C3)(10 P3) 86,400. 22. (15 P4)(4C3) 131,040. 23. (14C7)(15 P3) 9,369,360. 8 judges to a docket of criminal cases. Five of the judges are male and three are female. 25. Does the selection of judges involve a permutation or a combination? combination 26. In how many ways could three judges be chosen? 56 27. If the judges are chosen randomly, what is the probability that all 3 judges are male?. 5 , or about 18% 28 ____________ PERIOD _____. Reading 14-2 Readingto to Learn Learn Mathematics Mathematics, p. 841 Permutations and Combinations. 53. The coach must decide in which order the four swimmers should swim. He timed the swimmers in each possible order and chose the best time. How many relays did the four swimmers have to swim so that the coach could collect all of the data necessary? 24. Include the following in your answer: • a few sentences explaining why forming a Senate committee is a combination, and • an explanation of how to find the number of ways to select the committee if committee positions are based upon seniority.. JUDICIAL PROCEDURE The court system in a community needs to assign 3 out of. Pre-Activity. 52. How many different teams can he form? 15. How can combinations be used to form committees?. 24. SPORT In how many orders can the top five finishers in a race finish? 120. NAME ______________________________________________ DATE. 51. Assuming that each arrangement has an equal chance of occurring, what is the probability that she will form the words tap shoe on her first try? 1 30,240 SWIMMING For Exercises 52–54, use the following information. A swimming coach plans to pick four swimmers out of a group of 6 to form the 400-meter freestyle relay team.. 54. If Tomás is chosen to be on the team, what is the probability that he will swim in the third leg? 1 25% 4 WRITING IN MATH 55. Answer the question that was posed at the beginning of the lesson. See margin.. Evaluate each expression. 6. 11P3 990. ELL. Standardized Test Practice. How can combinations be used to form committees? Read the introduction to Lesson 14-2 at the top of page 760 in your textbook.. 56. There are 12 songs on a CD. If 10 songs are played randomly and each song is played once, how many arrangements are there? B A. 479,001,600. What is meant by the term combination?. A group of objects not arranged in any particular order. Complete the chart. Permutation or Combination?. Situation. Explain Your Choice. combination. See students’ work.. permutation. See students’ work.. 3. 4-digit student I.D. numbers. permutation. See students’ work.. 4. choosing 4 out of. combination. See students’ work.. A. B. 4. to go to a job fair. 766 2. arrangement of student. Chapter 14 Probability. work for the school art show. NAME ______________________________________________ DATE. 12 possible pizza toppings. Helping You Remember 5. To help you remember how the terms permutation and combination are different, think of everyday words that start with the letters P and C and that illustrate the meaning of each word. Explain how the words illustrate the two terms.. Sample answer: P for phone number, since the order of the digits in a phone number is important; C for club, since who is in a club is the important thing, not the order in which the names of the club members are listed. 14-2 Enrichment Enrichment,. Chapter 14 Probability. ____________ PERIOD _____. p. 842. Latin Squares In designing a statistical experiment, it is important to try to randomize the variables. For example, suppose 4 different motor oils are being compared to see which give the best gasoline mileage. An experimenter might then choose 4 different drivers and four different cars. To test-drive all the possible combinations, the experimenter would need 64 test-drives. To reduce the number of test drives, a statistician might use an arrangement called a Latin Square.. D1. D2. D3. D4. C1. A. B. C. D. For this example, the four motor oils are labeled A, B, C, and D and are arranged as shown. Each oil must appear exactly one time in each row and column of the square.. C2. B. A. D. C. C3. C. D. A. B. The drivers are labeled D1, D2, D3, and D4; the cars are labeled C1, C2, C3, and C4.. C4. D. C. B. A. Now, the number of test-drives is just 16, one for each cell of the Latin Square.. 766. B. 239,500,800. C. 66. D. 1. 57. Julie remembered that the 4 digits of her locker combination were 4, 9, 15, and 22, but not their order. What is the maximum number of attempts Julie could make before her locker opened? C. Reading the Lesson. 1. 3 of 7 students are chosen. Beverage e Lemonad Milk Soft Drink. 47. Find the number of possible meal combinations. 36. b. What is the probability that the first two digits are odd numbers with the other digits any of the remaining numbers?. n! (n r)!. n Pr. Side Dish Soup Salad French Fries. Lesson 14-2. Example 1 n Pr. n! (n r)!. . 16. C. 24. D. 256.

No results found