[Math] Need a thorough explanation of this combination problem

combinatorics

From a group of 5 women and 7 men, how many different committees consisting of 2 women and 3 men can be formed?

This one's easy. There's two experiments (ex 1 = committees of men)(ex 2 = committees of women) so it's just $5 \choose 2$$7 \choose 3$.

But the next question is

What if 2 of the men are feuding and refuse to serve on the committee together?

I don't understand this question at all.

Best Answer

As counterpoint to Ross's forward solution, here is a backward solution. You've already counted all the committees, now let's subtract the "bad" ones, where two feuding men are serving together. You also need two women and a non-feuding man, so there are $${5\choose 2}{5\choose 1}$$ bad committees, which you can subtract from $${5\choose 2}{7\choose 3}$$ to find your answer.