**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 ^{n}H_{r} = ^{n+r-1}C_{r} .

**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 3^{15} 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 – 1}C_{15} = ^{17}C_{2} ways (n = 3, r = 5).

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