Consider all n! permutations of the set {1, 2, 3, . . . , n}. We say that a permutation has i fixed points if i elements appear in their respective position. For example, if n = 3, the permutation {2,3,1} has zero fixed points, {3,2,1} has one fixed point, and {1, 2, 3} has three fixed points. Let pn(k) be the number of permutations of {1, 2, 3, . . . , n} that have exactly k fixed points. For example, we have p3(0) = 2, p3(1) = 3, p3(2) = 0, and p3(3) = 1. (a) Suppose n = 4. Determine the values of p4(0), p4(1), p4(2), p4(3), and p4(4). (b) Suppose n = 10. Determine the values of p10(10), p10(9), and p10(8). Clearly justify each answer. (c) Prove that 0·pn(0)+1·pn(1)+2·pn(2)+3·pn(3)+…+n·pn(n) = n! for all positive integers n. Hint: You can verify this relation for n=3 (d) There are n friends who eat out at a fancy restaurant, with each person ordering a unique appetizer. Unfortunately, the server at this super-expensive restaurant didn’t write down who ordered which dish. To avoid embarrassment, the server randomly placed an appetizer in front of each person, hoping that they guessed correctly. Assume all n! permutations are equally likely. Use part (c) to determine the expected value of the number of people who received the appetizer they ordered.
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