Consider the following method of shuffling a deck of n playing cards, numbered 1 through n. Take the top card from the deck and then replace it so that it is equally likely to be put under exactly k cards, for k = 0, 1, . , n - 1. Continue doing this operation until the card that was initially on the bottom of the deck is now on top. Then do it one more time and stop
(a) Suppose that at some point there are k cards beneath the one that was originally on the bottom of the deck Given this set of k cards explain why each of the possible k! orderings is equally likely to be the ordering of the last k cards
(b) Conclude that the final ordering of the deck is equally likely to be any of the N! possible orderings.
(c) Find the expected number of times the shuffling operation is per-formed.
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