## The Number of Ways to Shuffle Cards Clumsily

I am working on a combinatorial problem.  My current approach is to “shuffle together” the solutions to two smaller problems.  Hence the following.

Problem:  How many ways are there to clumsily shuffle two piles of cards together?  Clumsy shuffles include those where cards in one or both piles stick to each other.  This graphic explains.

Clumsy Shuffle Examples

The problem I was working on caused me to reason in this way.  Let’s shuffle two piles of four cards each together.  The total number of ways to order eight distinct cards is $8!$.  Each pile of cards is in a distinct order – one of $4!$ possibilities.  So the number of ways to shuffle the cards must be $\frac{8!}{4!4!}$ since there are $4!4!$ possible orders of the two piles.

But wait.  This is just $8C4$, the number of ways to choose four objects from eight objects.  Thus this more enlightened rationale:  We have eight slots into which to place the cards. Pick four of those slots and put the cards from the first pile in the slots in order.  Put the cards from the other pile in order in the remaining slots.  The number of ways to do this is $8C4$.  This diagram provides a visualization.

Shuffle – 8 Choose 4