70. Let N denote the number of cycles that result in the match problem. (a) Let Mn...

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70. Let N denote the number of cycles that result in the match problem.

(a) Let Mn = E[N], and derive an equation for Mn in terms of M1,...,Mn−1.

(b) Let Cj denote the size of the cycle that contains person j . Argue that N =n j=1 1/Cj and use the preceding to determine E[N].

(c) Find the probability that persons 1, 2,...,k are all in the same cycle.

(d) Find the probability that 1, 2,...,k is a cycle.

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