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$1$ $\underline{\text{Card Shuffling}}$  \vspace{2mm}  In discrete mathematics, a riffle shuffle is a shuffling of cards in which the top half is placed in one hand the the other half in the opposite hand. The cards are then alternatively interlaced with one another. There are two different types of riffle shuffles that will be the topic of discussion: the out-shuffle and the in-shuffle. An out-shuffle keeps the top card on top and the bottom card on bottom. When the top card is, instead, placed in the second position we have an in-shuffle.