A committee of 12 members is to be formed from 9 women and 8 men. The number of committees in which the…
A committee of 12 members is to be formed from 9 women and 8 men. The number of committees in which the women are in majority is
$2720$
$2702$
$2270$
$2278$
Solution
A committee of 12 members is to be formed when women are in majority.
Case I 9 women and 3 men
$\begin{aligned} \therefore \text { Number of ways }={ }^9 C_9 \times{ }^8 C_3 & =1 \times \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \\ & =56\end{aligned}$
Case II 8 women and 4 men
$\therefore$ Number of ways $={ }^9 C_8 \times{ }^8 C_4$
$\begin{aligned} & =9 \times \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} \\ & =630\end{aligned}$
Case III 7 women and 5 men
$\therefore$ Number of ways $={ }^9 C_7 \times{ }^8 C_5$
$\begin{aligned} & =\frac{9 \times 8}{2 \times 1} \times \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \\ & =36 \times 56=2016\end{aligned}$
$\begin{aligned} \therefore \text { Required number of ways } & =56+630+2016 \\ & =2702\end{aligned}$