From a group of 6 men and 4 women a Committee of 4 persons is to be formed. In how many different ways can it be done, so that the committee has at least 2 men?

Question: From a group of 6 men and 4 women a Committee of 4 persons is to be formed. In how many different ways can it be done, so that the committee has at least 2 men?

Solution:
The committee of 4 persons is to be so formed that it has at least 2 men. The different ways that we can choose to form such a committee are:
2m. 2w in = ⁶C₂ × ⁴C₂ = 90
3m. 1w in = ⁶C₃ × ⁴C₁ = 80
4m in = ⁶C₄ = 15

∴ Total no. of different ways in which a committee of 4 persons can be formed so that it has at least 2 men = 90 + 18 + 15
= 185