Permutations and Combinations Summary

• 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.

• 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 ⁿP_r or by P(n, r).

• For 1 £ 4 £ n, ⁿP_r = n(n - 1)(n - 2)...... r factors.

          In particular, ⁿP_n = 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+.......+p_r 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 n^r.

• 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  ⁿC_r or by C(n, r).

• In particular, ⁿC₀ = ⁿC_n = 1.

• If 1 £ r £ n, then ⁿC_r = ⁿC_n-r.

• If 1 £ r £ n, then ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r.