Question

A committee of $12$ persons is to be formed from $9$ women and $8$ men in which at least $5$ women have to be included in a committee. Then the number of committees in which the women are in majority and men are in majority are respectively

• A. $4784,1008$
• B. $2702,3360$
• C. $6062,2702$
• D. $2702,1008$

Answer 1

The correct option is D

$2702,1008$

Explanation for the correct option:

Step 1: Find ways to select at least $5$ women in a committee of $12$ persons when women are the majority:

Women will be in the majority if the number of women will be $7,8,9$.

$\begin{array}{rcl}n& =& {}^{9}C_7·{}^{8}C_5+{}^{9}C_8·{}^{8}C_4+{}^{9}C_9·{}^{8}C_3\\ & =& \frac{9·8}{2}·\frac{8·7·6}{3·2}+9·\frac{8·7·6·5}{4·3·2}+1·\frac{8·7·6}{3·2}\\ & =& 2016+630+56\\ & =& 2702\end{array}$

Step 2: Find ways to select at least $5$ women in a committee of $12$ persons when men are the majority:

Men will be in the majority if the number of Men will be $7$.

$\begin{array}{rcl}n& =& {}^{9}C_5·{}^{8}C_7\\ & =& \frac{9·8·7·6}{4·3·2}·8\\ & =& 1008\end{array}$

Hence, option (D) is the correct .