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# Test of Mathematics Solution Subjective 48 - The Gifts Distribution

This is a Test of Mathematics Solution Subjective 48 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

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## Problem

Find the different number of ways $5$ different gifts can be presented to $3$ children so that each child receives at least one gift.

## Solution:

There are two possible ways in which the gifts can be distributed.

Case 1: They are distributed as $2,2,1$.

So first we choose the children who get $2$ gifts each in $^3C_2$ ways. Then we choose the gifts in $\frac{5!}{2!.2!}$ ways.

Thus total number of ways = $3.\frac{5!}{2!2!}= 90$ ways.

Case 2: They are distributed as $3,1,1$.

So first we choose the child who gets $3$ gifts  in $^3C_1$ ways. Then we choose the gifts in $\frac{5!}{3!}$ ways.

Thus total number of ways = $3.\frac{5!}{3!}= 60$ ways.

Therefore total number of ways to distribute the gifts = $90+60$ = $150$ ways.