Twenty people work at a small startup. A six-person committee is to be formed.
(a) Let D denote the set of all possible committees. Find |D|.
(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|.
(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|.
(d) Find |P ? Q|.
(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.
(f) Find |S|.
(g) How many committees can we form with no unhappy employees?
(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?
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