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 2 of the dice land on six. Then the other die will be thrown, and if it lands on six, then N = 2.) Let m, =E[N].

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

(b) Let X, denote the number of dice rolled on the ith throw. Find 1 .