52 Factorial - Eli5: The Playing Card
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Garlic Stir-Fried Spinach From Lucas Sin
A foolproof way lớn sauté bright green, toothsome leafy greens that have both flavor và textural integrity.This Thanksgiving I was hanging out with my little cousin, who was dealing cards from a standard 52 thẻ deck. I asked her the question:
“How many times could you giảm giá khuyến mãi the whole deck & see a unique order of cards — different than every other time you dealt the deck in the past?”
Or lớn 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 khổng lồ try khổng lồ grok that number online, and got variations of the same answer, which were pretty un-explainable to my cousin. So instead I’ve decided lớn make a try myself.
Imagine you stopped everything you were doing, and for the rest of your life, every second of your life, night & 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 lớn know how many unique giao dịch there are, and have a deep distrust of mathematicians — you need to lớn see it with your own eyes.
Just one of the many unique deals (“permutations”)You first have a hand that happens to lớn start with the Jack of Clubs, then the Jack of Hearts, then the Jack of Spades & then the 49 other cards. “Hmm, interesting one!” You think… khổng lồ 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 continues with the 51 other cards.
You keep going, & by the next week you’re pretty bored. When am I going to lớn be done? That Tuesday you see a khuyễn mãi giảm giá which has the Jack of Hearts, then the Jack of Clubs, then the Jack of Spades và 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 khổng lồ a room full of empty pizza boxes, và at this point imaginary friends as you haven’t slept in a week.
Don’t try this at homeYou realize you’ll need some help to lớn keep doing this. So you manage khổng lồ convince every other person on earth, all 7 Billion of them, to vì chưng it with you, every second of every day. You must figure this out! For some reason you aren’t able lớn convince the mathematicians lớn work on your project, but again you’ve never been big fans of them anyways.
Not even close to 7 billion
Still, things aren’t progressing fast enough. You decide lớn invent a time machine, and a Fountain of Youth, and everyone goes back lớn 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 lớn see dinosaurs! There are a lot of seconds between now & the beginning of the universe — you asked an Astronomer và they told you there were 4×10²⁷ seconds since then, or 4 followed by 17 zeros. The mathematicians come with you, but still refuse lớn help giảm giá khuyến mãi cards. They’re just in it khổng lồ see dinosaurs, they say. One shouts: “Good luck with your little project!” và they all giggle.
This makes you a bit uneasy, but you don’t have time for their antics, và quickly forget them as you return to dealing cards.
All 7 billion people, all working every second of every day from the Big Bang onwards…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 & 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 chất lượng 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 và convincing everyone to lớn join you. Time keeps passing và people keep dealing cards until we arrive back at today’s date.
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You look back at all the progress you’ve made. Wow! We’ve been able to do 3×10²⁷th combinations, that’s amazing! That’s a 3 followed by 27 zeros. So many different giao dịch of the deck, all unique.
“We must be close!”, you say confidently to lớn 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 hotline these ‘permutations’).
The smartest & most kind mathematician taps you on the shoulder. You asked her to lớn chart the progress you’ve been making towards collecting all the different possibilities, since she didn’t seem to lớn giggle as hard as the others when she walked by & 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.
Technically it’s 0.000000000000000000000000000000000000001% complete. You are still furious.“We have to lớn time travel again to lớn the beginning of the universe… you need more time khổng lồ finish…” She cautiously suggests.
“Okay, well, fine. How many more times?” You shoot back, still furious.
“10⁴⁰ 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 khổng lồ 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 lớn travel back in time this many times”, she repeats confidently, with a cảnh báo 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 lớn 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 8×10⁶⁷, 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 & looking strangely at you. You don’t like their look. As they start standing up & moving towards you, you run towards your time machine, setting it khổng lồ transport only yourself khổng lồ the Jurassic period.
In reality, it’s far worse: you don’t get the consolation prize of being Chris Pratt as you get eaten by dinosaurs.“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 lớn take your chances with the terrible lizards instead.
In the background, the mathematicians start giggling again.
