Math - Probability

Why Is Every Shuffled Card Order Unique?

A standard deck has 52 cards. Shuffle it well and you get a particular order. The question is: how many different orders are even possible?

52! = 52 × 51 × 50 × ... × 2 × 1 Read as "52 factorial" — the exclamation mark means you multiply all numbers from 52 down to 1

💬 Mia asks Grandpa Theo

Mia

52 times 51 times 50 and so on - that can't possibly give THAT many possibilities, can it?

Grandpa Theo

Oh, it can - that's exactly the trick with factorials, they explode incredibly fast! For the first card you have 52 options, for the second only 51 left (one is already taken), for the third 50, and so on down to the last card.

Mia

Okay, show me with fewer cards so I can follow along.

Grandpa Theo

Sure! With 3 cards there are 3! = 6 orders. With 4 cards already 4! = 24. With 5 cards, 5! = 120. See how it grows? Not slowly, but explosively - and with 52 cards you end up with a number that has 68 digits.

The number written out

52! ≈ 8.07 × 10⁶⁷ — that's an 8 followed by 67 zeros (more precisely: 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000).

For comparison: the universe has existed for about 4.35 × 10¹⁷ seconds since the Big Bang. Even if every one of the roughly 8 billion people on Earth had shuffled a new deck every single second since the Big Bang, that would only add up to about 3.5 × 10²⁷ orders - still 40 orders of magnitude short of 52!

So the next time you shuffle a deck well, the resulting order is, with near certainty, one that has never existed before in the history of humankind - and is unlikely to ever appear again.

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