Consider the following approach to shuffling a deck of n cards. Starting with any initial ordering of the cards, one of the numbers 1, 2,..., n is randomly chosen in such a manner that each one is equally likely to be selected. If number i is chosen, then we take the card that is in position i and put it on top of the deck—that is, we put that card in position 1. We then repeatedly perform the same operation.

Show that, in the limit, the deck is perfectly shuffled in the sense that the resultant ordering is equally likely to be any of the n! possible orderings.