Question

A committee of $12$ 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

$2702,1008$

An explanation for the correct answer:

Step 1: Find the possible number of the ways the committee.

We have been given that, a committee of $12$ is to be formed from $9$ women and $8$ men in which at least $5$ women have to be included in a committee.

We need to find the number of committees in which the women are in majority and men are in majority are respectively.

As a committee must be of $12$ people and is to be formed from $9$ women and $8$ .

Total number of ways ,

$$\begin{gathered} =({}^{9}C_5\times {}^{8}C_7)+({}^{9}C_6\times {}^{8}C_6)+({}^{9}C_7\times {}^{8}C_5)+({}^{9}C_8\times {}^{8}C_4)+({}^{9}C_9\times {}^{8}C_3) \\ =(\frac{9!}{5!(9-5)!}\times \frac{8!}{7!(8-7)!})+(\frac{9!}{6!(9-6)!}\times \frac{8!}{6!(8-6)!})+(\frac{9!}{7!(9-7)!}\times \frac{8!}{5!(8-5)!})+(\frac{9!}{8!(9-8)!}\times \frac{8!}{4!(8-4)!})+(\frac{9!}{9!(9-9)!}\times \frac{8!}{3!(8-3)!}) \\ =1008+2352+2016+630+56 \\ =6062 \end{gathered}$$

Step 2: Find the number of possible ways when the women are in majority.

$$\begin{gathered} =({}^{9}C_7\times {}^{8}C_5)+({}^{9}C_8\times {}^{8}C_4)+({}^{9}C_9\times {}^{8}C_3) \\ =(\frac{9!}{7!(9-7)!}\times \frac{8!}{5!(8-5)!})+(\frac{9!}{8!(9-8)!}\times \frac{8!}{4!(8-4)!})+(\frac{9!}{9!(9-9)!}\times \frac{8!}{3!(8-3)!}) \\ =2016+630+56 \\ =\mathbf{2702} \end{gathered}$$

Step 3: Find the number of possible ways when the men are in majority.

$$\begin{gathered} =({}^{9}C_5\times {}^{8}C_7) \\ =(\frac{9!}{5!(9-5)!}\times \frac{8!}{7!(8-7)!}) \\ =\mathbf{1008} \end{gathered}$$

Therefore, option (D) is the correct answer.