The fundamental principle of counting (F.P.C) states that if an operation can be performed in m different ways and if for each such choice, another operation can be performed in n different ways, then both operations, in succession can be performed in exactly mn different ways. The principle can also be generalized, for even more than two operations.
For n Î N,
the factorial of n is defined as n! = 1 ´
2 ´
3 ´ .....
´ n.
0! is defined as 1.
The arrangements of a number of things taking some or all of them at a time are called permutations. The total number of permutations
of n distinct things taking r(1 £ r
£ n) at a time is denoted by nPr or by P(n,
r).
For 1 £ 4 £ n,
nPr = n(n - 1)(n - 2)...... r factors.
In particular, nPn
= n(n - 1)(n - 2).....n factors.
= n(n - 1)(n - 2)......
3.2.1. = n!
If p1 objects are of first kind and p2 objects are of the second kind, then the total number of permutations of all the p1+p2 objects is
given by
If p1 objects are of the ith kind and i = 1,2,3,….r, then the total number of permutations of all the p1+p2+p3+.......+pr objects is
given by
The number of permutations of n different things taking r at a time when each thing is allowed to repeat any number of times in any arrangement is given by nr.
The number of circular permutations of n different things is given by (n - 1)!.
If the number of circular permutations of n different things when an anticlockwise circular permutation and its corresponding clockwise circular permutation are considered as same circular permutation,
then the number of circular permutations is
The selections (groups) of a number of things taking some or all of them at a time are called combinations. The total number of
combinations of n distinct things taking r(1£
r £ n) at a time is denoted by
nCr or by C(n, r).
