There are 52 cards in a regular deck of playing cards. There are a lot of various permutations those cards can take, which is why playing card games remains interesting. I just asked my ten-year-old daughter how many possible combinations she thinks you can make with 52 cards. Her answer? “A thousand.” I said, “Oh, it’s more than that.” She said, “A million? A trillion?” Then I showed her the answer. Here it is:
That’s how many ways you can arrange the 52 cards in a straight deck. It can also be written 8.066 x 10^67 (or 8.066e67). That is a big, big, big number. How do we know it so precisely? It’s actually a very easy formula: it’s 52 factorial, which is written 52! In case you’ve forgotten about factorials, here’s a quick factorial tutorial. (Oh God, sorry I wrote that.)
The factorial of an integer n is n times (n – 1) time (n – 2) and so forth until you get to 1. For example,
3! = 3 x 2 x 1 = 6
5! = 5 x 4 x 3 x 2 x 1 = 120
As you can guess, the numbers start to grow really, really quickly. Why is this formula the way to work out the number of permutations of a straight deck? Easy.
The first card in the deck can be any of 52 cards.
The second card can be any of the remaining cards, thus any of 51 cards.
The third card can be any of 50 cards.
And so forth…
The 51st card in the deck can be either of the 2 remaining cards.
The final, 52nd card has only one possibility, whatever card is left.
52 x 51 x 50 … x 2 x 1
also known as
Okay. Now how big a number is that? Really big. Extraordinarily big. Like monstrously big. Like if you could list ten million of the possible arrangements every second, you still wouldn’t be done once the sun went supernova. Like if you added up all the hydrogen atoms in the sun, and then did that again about 10 trillion times, you’d be close. But I really, really want to encourage you to read another article I found online about this. It’s an astonishing way to look at just how big this number is.
I know, right? Holy crap! I will never take shuffling for granted again. Or the Pacific Ocean, for that matter! (Don’t get the reference? Then you didn’t follow the link, did you?) Math is weird. Really, really weird. And probabilities are very, very big numbers. Probably.