Question

A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of exactly 3 women?

Answer 1

Number of men = 8

Number of women = 4

Number of peoples in the committee = 7

Exactly 3 women

In a 7 member committee, women must be 3. Therefore, the remaining 4 must be men.

The number of ways of selecting 3 women from 4 women = 4C₃

The number of ways of selecting 4 men from 8 men = 8C₄

∴ The total number of ways of selection is = 4C₃ × 8C₄ 

= ${\displaystyle \frac{4!}{3!(4-3)!}\times \frac{8!}{4!(8-4)!}}$

= ${\displaystyle \frac{4!}{3!\times 1!}\times \frac{8}{4!\times 4!}}$

= ${\displaystyle \frac{4\times 3!}{3!}\times \frac{8\times 7\times 6\times 5\times 4!}{4!\times 4!}}$

= ${\displaystyle 4\times \frac{8\times 7\times 6\times 5}{4!}}$

= ${\displaystyle \frac{4\times 8\times 7\times 6\times 5}{4\times 3\times 2\times 1}}$

= 8 × 7 × 5

= 280