Question

A committee of $7$ has to be formed from $9$ boys and $4$ girls. In how many ways can this be done when the committee consists of:
(i) exactly $3$ girls?
(ii) at least $3$ girls?
(iii) at most $3$ girls?

Answer 1

(i) exactly $3$ girls

This means you have to choose $3$ girls from $4$ and $4$ boys from $9$ to form a committee of $7$. This can be done in

${}^4{C}_3{\times }^9{C}_4$

(ii) At least $3$ girls

This means girls can either be $3$ or $4$

So first choose $3$ girls and $4$ boys or $4$ girls and $3$ boys

Total ways = ${}^4{C}_3{\times }^9{C}_4$ + ${}^4{C}_4{\times }^9{C}_3$

(iii) At most $3$ girls

This means we have to choose $1$ or $2$ or $3$ girls

So this can be done in

${}^4{C}_0{\times }^9{C}_7{+}^4{C}_1{\times }^9{C}_6$ + ${}^4{C}_2{\times }^9{C}_5$ + ${}^4{C}_3{\times }^9{C}_4$ ways.