Does anyone know how to apply Vandermonde's Identity to the following problem?

Seven women and nine men are on the faculty in the mathematics department at school.
a) How many ways are there to select a committee of five members if at least one woman must be on the committee?
b) How many ways are there to select a committee of five members of the department if at least one woman and one man must be on the committee?

To figure it out with regular combinations for a, I took the combination of all committees of five (4,368) and subtracted the possible committees of just men (126) to get the committes with at least 1 woman (4,242)

For b, I took the number of total committe combinations (4,368) and subtracted both committees with just men (126) and just women (21) to get 4,221 committees with at least one man and one woman.

How do you apply Vandermonde's identity to this problem? I thought it just allowed you to unite two sets and create combinations from those two WHOLE sets.