Question

A committee of $9$ people is to be chosen from a group of $8$ men and $7$ women. How many committes are possible if there are $5$ men and $4$ women in one committee?

• A. $1900$
• B. $1950$
• C. $1960$
• D. $2000$

Answer 1

The correct option is C $1960$

Given:
$\bullet$ Total number of men $=8$
$\bullet$ Total number of women $=7$

From $8$ men, we need to select $5$ men. This can be done in ${}^8{C}_5$ ways.

From $7$ women, we need to select $4$ women. This can be done in ${}^7{C}_4$ ways.

Number of committees which includes $5$ men and $4$ women. can be obtained by multiplying ${}^8{C}_5$ by ${}^7{C}_4$ as both the events should occur simultaneously.

Number of committees:
${}^8{C}_5\times {}^7{C}_4$
$=\frac{8!}{5!(8-5)!}\times \frac{7!}{4!(7-4)!}$

$=\frac{8!}{5!3!}\times \frac{7!}{4!3!}$

$=56\times 35$
$=1960$

The number of committees possible if there are $5$ men and $4$ women in one committee would be $1960$.

Therefore, option (c.) is the correct answer.