How large is 52!?

This Thanksgiving I was hanging out with my little cousin, who was dealing cards from a standard 52 card deck. I asked her the question:

“How many times could you deal the whole deck and see a unique order of cards – different than every other time you dealt the deck in the past?”

Or to put it another way,

“How many unique deals (‘permutations’) of all 52 cards are there?”

If you’ve done any number theory (or looked at the title of the post), you know the answer – 52! (Aka 52 factorial. I would be equally excited about 52 being the answer, however – that would be very strange and interesting)

I looked up ways to try to grok that number online, and got variations of the same answer, which were pretty un-explainable to my cousin. So instead I’ve decided to make a try myself.

Imagine you stopped everything you were doing, and for the rest of your life, every second of your life, night and day, you dealt a new, magically shuffled hand of all 52 cards. This magic makes it so that you’ll never get a hand that you’ve seen before. You want to know how many unique deals there are, and have a deep distrust of mathematicians – you need to see it with your own eyes.

You first have a hand that happens to start with the Jack of Clubs, then the Jack of Hearts, then the Jack of Spades and then the 49 other cards. “Hmm, interesting one!” You think… to yourself, as you probably don’t have many friends if this is your hobby.

The next second, you have a hand that starts with the Ace of Hearts, then has the 51 other cards.

You keep going, and by the next week you’re pretty bored. When am I going to be done? That Tuesday you see a deal which has the Jack of Hearts, then the Jack of Clubs, then the Jack of Spades and all the rest of the cards in the exact same position as you got in the first deal.

“Wow, that’s super interesting, and super close to the first hand, but still a new hand!” You exclaim to a room full of empty pizza boxes, and at this point imaginary friends as you haven’t slept in a week.

You realize you’ll need some help to keep doing this. So you manage to convince every other person on earth, all 7 Billion of them, to do it with you, every second of every day. You must figure this out! For some reason you aren’t able to convince the mathematicians to work on your project, but again you’ve never been big fans of them anyways.

Still, things aren’t progressing fast enough. You decide to invent a time machine, and a Fountain of Youth, and everyone goes back to the beginning of the universe, starting again with all 7 Billion people all dealing hands of cards, all day, every day, once a second per person. You think this will certainly work, and then you’ll get to see dinosaurs! There are a lot of seconds between now and the beginning of the universe – you asked an Astronomer and they told you there were 4×10^27 seconds since then, or 4 followed by 17 zeros. The mathematicians come with you, but still refuse to help deal cards. They’re just in it to see dinosaurs, they say. One shouts: “Good luck with your little project!” and they all giggle.

This makes you a bit uneasy, but you don’t have time for their antics, and quickly forget them as you return to dealing cards.

The Big Bang happens, and the universe expands. The galaxies are formed and the first stars are created and explode, seeding the cosmos with iron and other heavy elements. The solar system is formed, and ever so slowly the planets. Earth cools, an ocean forms and life is sparked. All the while, 7 billion people, every day, all day, every second, are dealing unique hands of cards. Dinosaurs roam the earth, then are wiped out. Many eons ago you’ve become the most hated person in existence for creating this project and convincing everyone to join you. Time keeps passing and people keep dealing cards until we arrive back at today’s date.

You look back at all the progress you’ve made. Wow! We’ve been able to do 3×10^27th combinations, that’s amazing! That’s a 3 followed by 27 zeros. So many different deals of the deck, all unique.

“We must be close!”, you say confidently to the nearest exhausted card-flipping humans. They look at you with pure hatred in their eyes. You wish that maybe you had just tried to ask a mathematician in the first place to figure out if it’s even possible to find all the unique deals (the mathematicians call these ‘permutations’).

The smartest and most kind mathematician taps you on the shoulder. You asked her to chart the progress you’ve been making towards collecting all the different possibilities, since she didn’t seem to giggle as hard as the others when she walked by and saw you dealing cards. She hands you a chart.

“What is this, some kind of sick joke?” You demand, enraged as she’s handed you a chart of a progress bar that’s blank.

“We have to time travel again to the beginning of the universe… you need more time to finish…” She cautiously suggests.

“Okay, well, fine. How many more times?” You shoot back, still furious.

“10^40 times…” There’s a pause as you stare blankly at her.

“Let me say the number”, she says, “I don’t think that scientific notation really hit home for you… It’s: Three, zero, zero, zero, zero, zero, zero… “

“Just write it down”, you cut her off, muttering under your breath about how you never really liked mathematicians anyways.

She writes it on a piece of paper, explaining: “You’ll have to continue this experiment with all 7 billion people, repeatedly returning to the beginning of the universe to get more seconds to deal cards.”

“You’ll have to travel back in time this many times”, she repeats confidently, with a note of pity in her voice.

The slip of paper says: “30,000,000,000,000,000,000,000,000,000,000,000,000,000 times”.

Your knees give out as you slump to the floor.

In the depths of despair, you look up at the mathematician. She smiles.

“You know, you could have just asked us mathematicians in the beginning. The number of combinations is 52!, which is just 52*51*50…*3*2*1. This equals about 8×10^67, or 8 with 67 zeroes after it.”

You nod, feeling relieved that you know the answer. Then you look up. You see 7 billion people, all of whom have just abandoned their work and looking strangely at you. You don’t like their look. As they start standing up and moving towards you, you run towards your time machine, setting it to transport only yourself to the Jurassic period.

“I should have trusted the mathematicians …. !” Your voice fades as you begin the time travel. You’re still not sure you understand how large numbers or probability works, but you think it’s wise to take your chances with the terrible lizards instead.

In the background, the mathematicians start giggling again.

Numbers:

  • Age of universe = 4×10^17
  • 7,000,000,000 = 7×10^9
  • 52! = 8×10^67
  • 52! / 7 Billion / age of universe in seconds = 3×10^40

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