If one deck of 52 cards can have 52! possible arrangements, how many can you get when you have two decks?

Question

If one deck of 52 cards can have 52! possible arrangements, how many can you get when you have two decks ?

Answer 1

You have now 104 cards, but they are paired. Assuming the two decks are identical, you have $\frac {104!}{2^{52}}$ combinations. Indeed there are $104!$ possible shuffles if the two decks are different, but since they are identical, for each pair of identical cards, you can swap them and you have the same shuffle.

Answer 2

If the decks must stay separate, then you have 52! arrangements in deck 2 for each of the 52! arrangements in deck 1, so there are $52! \times 52!$ total arrangements.

If the decks are combined, you just increase the operand of the factorial to the new card count, so there are $(52 + 52)!$ total arrangements.