Given a random permutation of the integers in the set {1, 2, 3, 4, 5}, let X equal the

QUESTION:

Given a random permutation of the integers in the set {1, 2, 3, 4, 5}, let X equal the number of integers that are in their natural position. The moment-generating function of X is M(t) = 44 120 + 45 120 et + 20 120 e2t + 10 120 e3t + 1 120 e5t . (a) Find the mean and variance of X. (b) Find the probability that at least one integer is in its natural position. (c) Draw a graph of the probability histogram of the pmf of X.