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Problem #3 INDIVIDUAL Consider all n! permutations of the set {1, 2, 3, .. . ,n}. We say that a number 2' is a xed point of a permutation if 3' appears in the 1th position. For example, if n = 3, the permutation {2,3,1} has zero xed points, {3,2,1} has one xed point, and {1,2,3} has three xed points. Let p1,,(k) be the number of permutations of {1, 2, 3, . . . ,n} that have exactly I: xed points. For example, we have 103(0) = 2, 313(1) = 3, 113(2) = 0, and 113(3) = 1. (a) Suppose n = 4. Determine the values of 114(0), p4(1), 104(2), 114(3), and 104(4). (b) Suppose n = 10. Determine the values of plg(10), 1110(9), and 1010(8). Clearly justify each answer. (c) Prove that 0-10,,(0) + 1-pn(1)+ 2 -p,,(2) +3 -pn(3) + . . . +11. -p,.,(n) = n! for all positive integers n. ((1) There are 11. 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. Determine the expected value of the number of people who received the appetizer they ordered