34. A set of n dice is thrown. All those that land on six are put aside, and the others are again thrown. This is repeated until all the dice have landed on six. Let N denote the number of throws needed. (For instance, suppose that n = 3 and that on the initial throw exactly two of the dice land on six. Then the other die will be thrown, and if it lands on six, then N = 2.) Let mn = E[N].

(a) Derive a recursive formula for mn and use it to calculate mi , i = 2, 3, 4 and to show that m5 ≈ 13.024.

(b) Let Xi denote the number of dice rolled on the ith throw. Find E[Ni

=1 Xi].