(1) Let f : [n] r [n] be a permutation. Axed point of f is an element x E [n] such that f(x) = x. Now consider random permutations of [n] and let X be the random variable which represents the number of xed points of a given permutation. {a} What is the probability that X = 0? {b} What is the probability thatX = n 2? (c) What is the probability that X = n 1? {d} What is the expectation of X? (Hint: As usual, express X as the sum of wellchosen indicator variables and use the linearity of expectation.) (e) Let tk be the number of permutations of [n] that have exactly I: fixed points. Prove that 23:\" ktk = n!. (Hint: Use parts {aHd) of the problem and consider two different expressions for E 00.) The density f (a ) = e (I-A)2/202 correspond to a V2TO X O exponential random variable O geometric random variable O standard normal random variable O Binomial random variable O uniform continuous random variable O Bernoulli random variable O normal random variable