Problem 18 A deck consists of '11 cards, numbered 1 through an. The deck is going to be shufed. Assume that all permutations of the n cards are equally likely. Let 1 g m1, m2, k1, kg 3 n. a) How many permutations are possible? b) In how many permutations does card number m1 falls in place number k1? c) What is the chance that card number 1711 falls in place number h? d) What is the chance that card number m1 falls in place number k1 and card number m2 falls in place number k2? e) For 1 g i g n, say that a match occurs at place 3' if card 3' falls in place i. Let M be the total number of matches. Fine E(M) and Var(M). [Hint: It is a good move to write M as the sum of indicators. If you have forgotten what those are, see Example D on page 129 of your text, or the section entitled The Method of Indicators starting on page 168 of Pitmans probability book. And dont forget the indicators, nor any dependence issues, when you calculate the variance] I What happens to the distribution of M as n ) 00'? You dont have to give a mathematical proof, but you should provide a convincing heuristic argument. And you should show that your limit distribution is consistent with your answers to part c)