It’s easy - calculate 54!, then divide by 53 and 54.

52! is the number of possible arrangements of a deck of cards.

To see why, let’s say you pick cards at random from one deck and add them to another, initially empty, deck. First card can be any of 52, second card can be any of 51 remaining, and so on until there is only one card left. So total number of possible arrangements is

To calculate 52! by hand, start with Stirling’s approximation for , then divide by 53 and 54:

Now you should see why we started with 54 rather than 52. The convenient coincidence that can be used to simplify calculations.

Finally, multiply numerator and denominator by 4 and use the fact that to get

The exact value is 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 or about .

Source: http://www.solipsys.co.uk/new/Calculating52FactorialByHand.html