1. Counting - Discrete math

7 woman and 9 man are on the faculty in the math department in the school.

(1) How many ways are there to select a committee of five members of the department if at least one woman must be on the same committee ?

(2) how many ways are there to select a committee of five members of the department if at least one woman and at least one man must be on the committee .

2. You are considering a set which has $16$ elements, and has a partition in two subsets $A$ and $B$ which have $7$ and $9$ elements.
For 1), what you want to know is the number of 5-element subsets of $E$ that have at least one element in $A$. That means the "first" element $x$ must be in $A$, you have $7$ choices, then every subset with $4$ elements in $A-\{x\}$ union $\{x\}$ is a solution.
So there are $7\times C^{4}_{15}$ possible committees.