* Fundamental Principle of counting
:-If an event can occur in m ways and corresponding to each way another event can occur in p ways and corresponding to them, a third event can occur in r ways, then the total number of occurances of the events is mpr.
* Factorial :- The Product of first n natural numbres is known as Factorial. It is denoted by n! or
n!n.(n 1).(n 2)...3.2.1
n! n.(n 1)! n(n 1)(n 2)!
0! 1
* Permutations (Arrangements)
:- A Permutation is an arrangement in a definite order of a number of distinct objects taking some or all at a time.
The number of linear permutations of n different objects taking r at a time where 1 r n r, nN, is denoted by nP .r
If repetitions of objects is not allowed and arrangement is linear, the arrangements also called a linear Permutation.
nP = n ( nr 1)(n 2)....( n r 1)
n r n!
P =
(nr)! n Pr n ( n1 )P( r1 )
nPn n ! nP1 n nP2 n ( n 1)
Number of permutations of n distinct objects taken r at a time with repetitions allowed is n .r
* Permutations when some of the objects are identical (alike or one kind)
:-If P objects are alike, 1 P objects are alike different from earlier ones...., 2 P objects are alike differentK from earlier ones and nP1P2....PKthen the number of are permutations of n things is
1 2 K
n!
P !P !....P !
* Circular Permutation
:-The number of ways of arranging n different objects on a circle is called the number of circular
permutations of n objects.
The number of circular permutations of n different things is (n 1)!
In circular permutation, anti-clockwise and clockwise order of arrangements are considered as distinct permutations .
If anti-clockwise and clockwise order of arrangements are not distinct then the number of circular permutations of n distinct items are (n 1)!
2
Ex. 1 : The number of permutations of 5 persons seated around the round table is (5-1) ! = 4! . Because with respect to the table, the clockwise and anti-clockwise arrangements are distinct.
Ex. 2 : Arrangements of beads, necklace, arrangements of flowers in a garland etc., then the number of circular permutations of n distinct items is (n 1)!
2
The number of all permutations of n different objects taken r at a time, when a particular object is to be always included in each arrangement is r.n 1 Pr 1
The number of all permutations of n different objects taken r at a time, when a particular object is never taken in each arrangement is n1 Pr
Number of all permutations of n different objects taken r at a time in which two specified objects always occur together is 2 !( r 1). n2Pr 2
Combination (selection) :- The number of ways of selecting r things out of n different things is called r combination number of n things and is denoted by
n
The product of n consecutive integers is divisible by n!
The number of ways of selecting one or more items from a group of n distinct items is 2n 1
The number of ways of selecting none, one or more items from a group of n distinct items is 2n
The number of ways of selecting r items out of n identical items is 1.
The number of ways of selecting one or more (at least one ) items out of n identical items is n.
The number of ways of selecting none, one or more items out of n identical items is n+1.
(Here 1 is added for the case in no item is selected from the set of n identical items.)
The number of ways of selecting none, m items of one kind, n items of another kind and p items of another kind out of m+n+p items is (m+1) (n+1) (p+1).
The number of ways of selecting at least one (one or more) items from a collection of m items of one kind, n items of another kind and p items of another kind is (m+1) (n+1) (p+1) -1.
(Here -1 is used for rejecting that one case in which no item is selected.)
The number of ways of selecting at least one (one or more) item of each kind from a collection of m items of one kind, n items of another kind and p items of another kind is mnp.
The total number of ways of selecting one or more items from p identical items of one kind, q identical items of another kind, r identical items of another kind and n different items is
(p 1) (q 1) (r 1) 2n 1
The number of ways in which a selection of at least one item can be made from a collection of n distinct items and m identical items is 2 (m 1) 1n
(here -1 is used for rejecting that one case in which no item is selected)
Number of ways in which m+n+p items can be divided into unequal groups containing m, n and p items is m n 1 m n 1 n p p
(m n p )!
C . C . C
m !n !p !
Number of ways to distribute m+n+p items among 3 persons in the group containing m, n and p items is
(m n p)!
m!n!p! 3!
(Here 3! Is for arranging the things in between 3 persons as no two persons are alike)
The number of ways in which mn different items can be divided equally into m groups, each containing n objects and the order of the groups is not important, is m
(m n)! 1 ( n !) m!
The number of ways in which mn different items can be distributed equally among m different persons is m
(m n)!
( n!)
The number of diagonals of n sided convex polygon is nC2- n = n(n 3), n 3 2
There are n points in the plane such that no three of them are in the same straight line, then the number of lines that can be formed by joining them is nC2
There are n points in the plane such that no three of them are in the same straight line, then the number of triangles that can be formed by joining them is nC3
There are n points in the plane such that no three of them are in the same straight line except m of them which are in same straight line then the number of lines that can be formed by joining them is nC2m C21
There are n points in the plane such that no three of them are in the same straight line except m of them which are in same straight line, then the number of triangles that can be formed by joining them is nC3mC3
If n points lie on a circle then the number of straight lines formed by joining them isnC2
If n points lie on a circle then the number of triangles formed by joining them is nC3
If n points lie on a circle then the number of quadrilateral formed by joining them is nC4