Math 25700: Extra Homework Problem, due WEDNESDAY September 30.

Recall that there are two riffle shuffles A and B obtained as follows: divide the deck into the top 26 and bottom 26 cards. Then interweve the two decks card by card. (There are two different shuffles depending on whether the top card from the bottom deck ends up on top, or the top card from the top deck ends up on top. If we denote the shuffles by A and B respectively, then we saw in class that A^52 = 1.

1. Prove that not every permutation of 52 cards can be obtained by some combination of riffle shuffles.

2*. How many of the 52! = 80658175170943878571660636856403766975289505440883277824000000000000 permutations of 52 cards can be obtained in this way?

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