Question

In how many ways can a man divide $7$ gifts among his $3$ children if the eldest is to receive $3$ gifts and the others $2$ each?

Short answer

The no. of ways are $210$.

Step-by-step solution

Step 1. Given information.

It is given that,

Total no. of gifts are $=7$

Total no. of children $=3$

Gifts to be given to eldest child$=3$

Gifts to be given to other two children$=2\text{each}$

Step 2. Find the no. of ways.

The no. of ways in which the eldest child will receive $3$gifts and other two children will receive $2$gifts each out of $7$gifts is

$$\begin{gathered} \left(\begin{array}{c}7\\ 3,2,2\end{array}\right)=\frac{7!}{3!2!2!} \\ =\frac{7\times 6\times 5\times 4\times 3!}{2\times 2\times 3!} \\ =7\times 6\times 5 \\ =210 \end{gathered}$$

Therefore, the no. of ways are$210$.