How Many Ways Can You Sort a Deck of Cards?

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Given a standard 52-card deck, how many possible arrangements of the cards within it are there? Another way to ask this is: If you shuffle a deck of cards, what are the possible number of shuffled permutations?

The answer is, predictably, a whole lot. But the astounding fact is that if you shuffle a deck of cards right now, it's likely that the order your deck is in has never occurred before in human history.

In fact, the number of deck combinations is so great that if a new deck arrangement were written out every second starting at the Big Bang (which, of course, couldn't have happened, because cards didn't exist...but still...), we would still be coming up with new deck arrangements today, and for millions of years in the future. Because playing cards in their current form have "only" existed for centuries, and the possibility space is so enormous, a unique shuffled order is extremely likely.

Check out this TED-Ed video for an explanation of why this is the case, and the specifics of the math involved.

Check out this TED-Ed page for more resources. I particularly enjoyed the bit under "Dig Deeper" discussing the anagram possibilities of the Harry Potter phrase "Tom Marvolo Riddle."