Combinations

Let us suppose there are four persons A, B, C and D (actual names may be used here) and we have to select three of them to be a part of a committee. In how many ways can we make this selection? For example, A, B, C is one possible choice. Here the order of selection is immaterial.

Thus A, B, C is the same as B, A, C or C, A, B as long as the same three persons are selected. Thus the possible . Combinations distinct choices or selections are A, B, C ; A, B, D ; A, C, D and B, C, D . We may thus conclude that there are ways of selecting people out of .

Each choice or selection is referred to as a combination of different objects taken at a time. Suppose two persons are to be selected from four persons. The possible choices are: A, B : A, C : A, D : B, C : B, D : C, D . Thus the number of combinations of different objects taken at a time is .

The number of combinations of n different objects taken r at a time is represented by n C r . From the above we may conclude that C = and C = . Now, C is the number of combinations of objects taken at a time. Note that in each combination, the three objects may be arranged in !

ways. Thus the total number of permutations of objects taken at a time is C × ! . This is also equal to P .

Hence P = C × ! . In general, this leads to an important relationship between permutations and combinations as, n P r = n C r × r ! .

Normally for any reader there may be a confusion between permutation and combination. The following table with an example may be helpful in clearing the confusion. S.No Description Permutation Combination Number of Number of What is a Arrangement or Listing of Selections or Grouping of objects objects If the ordering of objects If the ordering of objects Where to use matters does not matter Representation n P r n C r Examples In a The number of batting The number of teams game of cricket line up of players out consisting of players of the players out of players In a The number of ways of The number of ways of process of prize distributing distinct distributing identical distribution prizes prizes In a The number of ways of The number of ways of committee choosing a President and forming a committee of formation a Vice-President for a persons from committee of members members In a The number of ways of The number of ways of process of choosing out of choosing out of choosing objects distinct objects one after distinct objects another simutaneously Theorem . : The number of combinations of n distinct objects taken r at a time is given by n C r = n !