How many ways can a committee be formed from 4 men and 6 woman with:

c)5 people, and not all of the 3 O'Hara sisters can be on the committee.

The answer is C(10,5) - C(7,2)

I do not understand where the C(7,2) comes from. Can someone explain this?

I tried to break it up into committees of 5 people:

0 men - 5 woman ---> I can only choose out of 5 women since one sister of 3 sisters is out

1 men - 4 women --> choose out of 5 women

2 men - 3 women --> choose out of 5 women

3 men - 2 women--> choose out of 6 women, since all 3 sisters cannot be chosen anyway

4 men - 1 women --> choose out of 6 women

5 men - 0 women --> choose out of 6 women

and in the end I add up.

C(4,0)*C(5,5) + C(4,1)*C(5,4) + C(4,2)*C(5,3) + C(4,3)*C(6,2) + C(4,4)*C(6,1)

But I get a different number, so this must be wrong.

Thanks for any help