**Question:**

In how many ways can committee of 5 be made out of 6 men and 4 women, containing at least 2 women?

**Solution:**

We need to include at least 2 women. If we include 2 women in the committee, then a number of men is 3. The number of ways, 2 women can be selected out of 4 is $={ }^{4} \mathrm{C}_{2}=6$ The number of ways, 3 men can be selected out of 6 is $={ }^{6} \mathrm{C}_{3}=$ 20 So, the committee can be formed including 2 women in $(20 \times 6)=120$ ways. If we include 3 women in the committee, then a number of men is 2 . The number of ways, 3 women can be selected out of 4 is $={ }^{4} C_{3}=4$. The number of ways, 2 men can be selected out of 6 is $={ }^{6} \mathrm{C}_{2}=15 \mathrm{So}$, the committee can be formed including 3 women in $(15 \times 4)=60$ ways. Therefore, the total number of ways the committee can be formed is $=(120+60)=180$ ways.