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

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

• Hint: how many cards are in two decks? Commented Oct 13, 2017 at 0:56
• Do you consider cards of identical rank and suit to be indistinguishable? Commented Oct 13, 2017 at 0:59
• Yes, so an ace of spades looks exactly the same in either deck, and so on and so forth for every card. Commented Oct 13, 2017 at 1:23

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.
• @DarthRatus Think of it this way. Assume that both the ace of spades are different and calculate. You get $104!$, but in every arrangement you can swap the two ace of spades. So you only get $\frac{104!}{2}$ arrangements. But you have every card that's a duplicate, so you need to divide by two 52 times. So there you go. Commented Oct 13, 2017 at 1:54
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.