Shuffling Cards

52 cards – 2 of clubs to Ace of clubs, 2 of diamonds to Ace of diamonds, 2 of hearts to Ace of hearts, and 2 of spades to Ace of spades – are arranged in a deck. We shuffle them in the following manner:

• We take the top card and put in a random place inside the deck.
• Once we get to the King of spades and put it somewhere in the deck, we stop.

Show that this method shuffles the deck uniformly, i.e. every permutation has the same chance to appear.

Notice that at all times the cards below the King of spades are shuffled uniformly. Therefore at the end, after we put the King of spades in a random place inside the deck, the entire shuffle will be uniform as well.

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