## Deck Shuffling

Cool fact: If you thoroughly shuffle an ordinary deck of 52 playing cards, chances are that the resulting arrangement of cards has never existed before (via Reddit and Reddit).

Quoting from the math subreddit:

Proof — easy numeric comparison. There are 52! possible orderings of a deck, and I’m assuming all are equally likely after your shuffling. Let’s wildly overestimate and assume that every second since the universe was created, a million decks of cards were shuffled and someone looked through them. Thus fewer than 10^24 orderings have ever been seen.

But at an incredibly crude estimate, 52! is at least 10^42 * 10!; let’s underestimate that again wildly by 10^42. That means that chances of your ordering ever having come up previously are at most 1 in 10^18.

(Note, by the birthday paradox, the chances that there have been two identical orderings observed by two people in history are quite a bit higher — perhaps even feasibly likely; I haven’t calculated it. But we’re looking here at the probability that a given ordering matches one of the ones previously seen.)

Discussion here.

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