a committee of four is selected from a group of 22 people.
if there are 10 women and 12 men in the group, how many different committees
have a majority of men
Notice that I crossed out the 20 and put 22 so it would make sense.
10 women and 12 men is 22 people, not only 20 people.
The group of 4, to have a majority of men, would have either
Case 1. 3 men and 1 woman
or
Case 2. 4 men.
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In case 1 we have 12 men CHOOSE 3, that's 12C3 = 220 ways to choose the 3
men. For each of those 20 ways, we have 10 women CHOOSE 1, that's 10C1=10.
So, for case 1, there are 220×10 = 2200.
In case 2 we have 12 men CHOOSE 4, that's 12C4 = 495 ways to choose the 4
men. So for case 2, there are 495.
Grand total for both cases = 2200+495 = 2695.
Edwin