Question: 69. In the match problem, say that (i,j),i < j , is a pair if i chooses j s hat and j chooses is hat.

69. In the match problem, say that (i,j),i < j , is a pair if i chooses j ’s hat and j chooses i’s hat.

(a) Find the expected number of pairs.

(b) Let Qn denote the probability that there are no pairs, and derive a recursive formula for Qn in terms of Qj ,j

Hint: Use the cycle concept.

(c) Use the recursion of part

(b) to find Q8.

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