This paper is about the following question: How many riffle shuffles mix adeck of card for games such as blackjack and bridge? An object that comes up inanswering this question is the descent polynomial associated with pairs ofdecks, where the decks are allowed to have repeated cards. We prove that theproblem of computing the descent polynomial given a pair of decks is$#P$-complete. We also prove that the coefficients of these polynomials can beapproximated using the bell curve. However, as must be expected in view of the$#P$-completeness result, approximations using the bell curve are not goodenough to answer our question. Some of our answers to the main question aresupported by theorems, and others are based on experiments supported byheuristic arguments. In the introduction, we carefully discuss the validity ofour answers.
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