Twenty people work at a small startup. A six-person committee is to be formed.

1. (a) Let D denote the set of all possible committees. Find |D|.

2. (b) Two of the workers, Alice and Beth, will be unhappy if they have to serve together. Let P be the set of possible committees with both Alice and Beth. Find |P|.

3. (c) Beth will also be unhappy if she has to serve with both Frank and Grace. Let Q be the set of possible committees on which Beth, Frank, and Grace would all serve together. Find |Q|.

4. (d) Find |P ? Q|.

5. (e) Let S be the set of all possible committees on which there is at least

   one unhappy employee. Express S in terms of only P and Q.

6. (f) Find |S|.

7. (g) How many committees can we form with no unhappy employees?

8. (h) Instead of one committee, we decide to form two six-person commit- tees. Each employee can be on at most one committee. How many ways can we form these committees, if employee happiness is nottaken into consideration?