Foundations of Computer Science Discrete Math A club with 20 women and 17 men ne

Question

Foundations of Computer Science
Discrete Math

A club with 20 women and 17 men needs to form a committee of size six.

- How many committees are possible?
- How many committees are possible if the committee must have three women and three men?
- How many committees are possible if the committee must have at least two men?
- How many committees are possible if the committee must consists of all women or all men?
- How many committees are possible if the committee must consists of all women or all men?

Explanation / Answer

How many committees are possible?

Possible committees: 37C6. Selecting 6 members from 37 available members.
- How many committees are possible if the committee must have three women and three men? 20C3 * 17C3. Selecting 3 members from the available 20 women, and selecting the remaining 3 members from available 17 men.
- How many committees are possible if the committee must have at least two men? 17C2 * 35C4 Selecting 2 men from available 17 men. And selecting the remaining 4 members from the other 20 + 15 = 35 people.
- How many committees are possible if the committee must consists of all women or all men? 17C6 + 20C6. This is about 2 independent sets. So, we are adding. Selecting all 6 members from the available 20 women, OR, selecting all 6 members from the available 17 men.
- How many committees are possible if the committee must consists of all women or all men?

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