A combination is a selection made by taking some or all of a number of objects, irrespective of their arrangements. The number of all combinations of n distinct objects, taken r at a time is denoted by ncr or c(n, r).
The number of combination of n different things taken r at a time is given by ncr = (n!)/[r!(n – r)!], where r is greater than or equal to 0 and less than or equal to n.
Following are the properties of combination:
1. nc0 = ncn = 1
2. ncr = ncn – r
3. ncr + ncr – 1 = n + 1cr
Combination when all objects are different: Each thing may be disposed of in two ways. It may either be included or rejected. Therefore, the total number of ways of disposing of all the things = 2 x 2 x 2 x … n times = 2n
But this includes the case in which all the things are rejected. Hence, the total number of ways in which one or more things are taken = 2n – 1.
Combination when all things are not different: Suppose, out of (p + q + r +…) things, p are alike of one kind, q are alike of a second kind, r alike of a third kind, and the rest different. Out of p things, we may take 0, 1, 2, 3…or p. Hence, they may be disposed of in (p + 1) ways. Similarly, q alike things may be disposed of in (q + 1) and r alike things in (r + 1) ways. The t different things may be disposed of in 2t ways. This includes that case in which all are rejected. Therefore, the total number of selections
= (p + 1)(q + 1)(r + 1)2t – 1.
Division into groups: To find the number of ways in which p + q things can be divided into two groups containing p and q things respectively. Each time when a set of p things is taken, a second set of q things is left behind. Hence, the required number of ways
= number of combinations of (p + q) things taking p at a time
= (p +q)Cq
= (p + q)!/p!.q!
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