The first part of the question I know how to answer:
How many committees of five men and four women can be formed from and organization with 43 women and 47 men?
The answer to this of course would be $C(47,5)\cdot C(43,4)=\frac{47!43!}{42!39!5!4!}$
The second part of the problem I'm having difficulty with:
Suppose in the previous scenario that among the members there are 20 married couples and there is a rule prohibiting spouses from serving on the same committee. How many five-man, four-woman committees can be formed under this new restriction?
My intuition tells me that there could at most be 4 married couples on the same committee with one extra man under the first part of the problem and I should be able to subtract all the cases with 4 couples, 3 couples, 2 couples and 1 couple on a committee. I'm not sure how I should write that mathematically since I'm uncertain if I'm choosing out of the 20 couples at that point or out of the entire group of people.