## Ways to shuffle a deck

I’ve seen the trivia fact pop up a few times recently that if you randomly shuffle a deck of cards, chances are the order they are in has never been encountered before in history. Seems a bit surprising, so let’s run some numbers.

If you have a deck of 52 cards, and you randomly pick one card from the pile to begin your new shuffled pile, there are 52 ways in which the new pile can begin (any one card can be picked). Now there are 51 cards left in the unshuffled pile, so the next random card you pick will have 51 different ways of manifesting itself. Since you can trace any history of cards (say 2♦ followed by J♥ or 7♠ followed by A♥ or…). That is, the number of ways you can build a new pile of two cards is

$\displaystyle N_2 = 52 \times 51$.

There are 50 cards left in the deck, so the third card would contribute a 50 to the multiplication, the fourth card would contribute a 49, etc, so that once you randomly selected all 52 cards the number of ways of forming the deck would be

$\displaystyle N_{52} = 52 \times 51 \times 50 \times \ldots \times 3 \times 2 \times 1 = 52!$

where I’ve used the symbol ! to denote multiplying the associated numbers by all whole numbers equal to or less than it down to one, which is called a factorial (so N! is read N factorial). The first few factorials are easy enough to compute in your head, but the soon begin to grow enormously is size (in fact, it grows faster than most commonly encountered functions). When you do run the numbers, necessarily with the help of a computer, you arrive at

$\displaystyle 52! \approx 8.07 \times 10^{67} \mbox{.}$

This value is far, far more than the number of molecules in the deck of cards, which is already enormous. It’s more than the number of stars in the universe. It’s safe to the say that of the few million people over the past few centuries who have assembled decks between tens and tens of thousands of times each, it’s nowhere near such an enormous number.

So provided you perform some good shuffles, the chances that your deck is in an order never before experienced seems perfectly valid.