COMBINATIONS

Meaning of combination is selection of objects.

Selection of Objects without Repetition:

The number of selections (combinations or groups) that can be formed from n different objects taken r (0 £ r £ n) at a time is:

Selection of objects with repetition:

The number of combinations of n distinct objects, taken r at a time when each may occur once, twice, thrice,….. Upto r times in any combination is nHr = n+r-1Cr .

Illustration 1:

Let 15 toys be distributed among 3 children subject to the condition that any child can take any number of toys. Find the required number of ways to do this if

(i)        Toys are distinct.                      (ii)       Toys are identical.

Solution:

(i) Toys are distinct

Here we have 3 children and we want the 15 toys to be distributed to the 3 children with repetition. In other words, it is same as selecting and arranging children 15 times out of 3 children with the condition that any child can be selected any no. of time, which can be done in 315 ways (n = 3, r = 15).

(ii) Toys are identical

Here we only have to select children 15 times out of 3 children with the condition that any child can be selected any number of times which can be done in               3 + 15 – 1C15 = 17C2 ways (n = 3, r = 5).