# Thread: Counting - Discrete math

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