Question

43. Selecting a Committee There are 7 women and 5 men in a department. How many ways can a committee of 4 people be selected? How many ways can this committee be selected if there must be 2 men and 2 women on the committee? How many ways can this committee be selected if there must be at least 2 women on the committee?

Answer 1

7 women and 5 men in a department

4 people is to be selected.

nCr n! r!* (n-r)!

1) A commitee must contain 2 men and 2 women

Total number of ways :

2 men selected from 5 men

and 2 women selected from 7 men

By multiplication principle,

= 5C2 * 7C2

= 10 * 21

= 210

210 ways commitee be selected if there must be 2 men and 2 women.

2)

A commitee contain atleast 2 women.

Cases -

i) A commitee contain 2 women and 2 men = 7C2 * 5C2 = 10 * 21 = 210

ii) A commitee contain 3 women and 1 men = 7C3 * 5C1 =35 * 5 = 175

iii) A commitee contain 4 women and 0 men = 7C4 * 5C0 = 35 * 1= 35

By Addition principle

Total number of ways commitee be selected = 210 + 175 + 35 = 420