Question: 29. Balls numbered 1,2, . . . , and n are randomly placed into cells numbered 1, 2, . . . , and n. Therefore,

29. Balls numbered 1,2, . . . , and n are randomly placed into cells numbered 1, 2, . . . , and n. Therefore, for 1 ≤ i ≤ n and 1 ≤ j ≤ n, the probability that ball i is in cell j is 1/n. For each i, 1 ≤ i ≤ n, if ball i is in cell i, we say that a match has occurred at cell i.

(a) What is the probability of exactly k matches?

(b) Let n → ∞. Show that the probability mass function of the number of matches is Poisson with mean 1.

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