How many ways can you arrange a deck of cards? – Yannay Khaikin


Pick a card, any card. Actually, just pick up all of them and take a look. This standard 52-card deck has been used for centuries. Everyday, thousands just like it are shuffled in casinos all over the world, the order rearranged each time. And yet, every time you pick up a well-shuffled deck like this one, you are almost certainly holding an arrangement of cards that has never before existed in all of history. How can this be? The answer lies in how many different arrangements of 52 cards, or any objects, are possible. Now, 52 may not seem like such a high number, but let’s start with an even smaller one. Say we have four people trying to sit in four numbered chairs. How many ways can they be seated? To start off, any of the four people can sit in the first chair. One this choice is made, only three people remain standing. After the second person sits down, only two people are left as candidates for the third chair. And after the third person has sat down, the last person standing has no choice but to sit in the fourth chair. If we manually write out all the possible arrangements, or permutations, it turns out that there are 24 ways that four people can be seated into four chairs, but when dealing with larger numbers, this can take quite a while. So let’s see if there’s a quicker way. Going from the beginning again, you can see that each of the four initial choices for the first chair leads to three more possible choices for the second chair, and each of those choices leads to two more for the third chair. So instead of counting each final scenario individually, we can multiply the number of choices for each chair: four times three times two times one to achieve the same result of 24. An interesting pattern emerges. We start with the number of objects we’re arranging, four in this case, and multiply it by consecutively smaller integers until we reach one. This is an exciting discovery. So exciting that mathematicians have chosen to symbolize this kind of calculation, known as a factorial, with an exclamation mark. As a general rule, the factorial of any positive integer is calculated as the product of that same integer and all smaller integers down to one. In our simple example, the number of ways four people can be arranged into chairs is written as four factorial, which equals 24. So let’s go back to our deck. Just as there were four factorial ways of arranging four people, there are 52 factorial ways of arranging 52 cards. Fortunately, we don’t have to calculate this by hand. Just enter the function into a calculator, and it will show you that the number of possible arrangements is 8.07 x 10^67, or roughly eight followed by 67 zeros. Just how big is this number? Well, if a new permutation of 52 cards were written out every second starting 13.8 billion years ago, when the Big Bang is thought to have occurred, the writing would still be continuing today and for millions of years to come. In fact, there are more possible ways to arrange this simple deck of cards than there are atoms on Earth. So the next time it’s your turn to shuffle, take a moment to remember that you’re holding something that may have never before existed and may never exist again.

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Reader Comments

  1. Abandoned

    the number is 80658175170943878571660636856403766975289505440883277824000000000000,

    btw factorial of 1000 is 402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

  2. OofingOofs

    so when you arrange the cards you will get 1 possibility out of 80000000000000000000000000000000000000000000000000000000000000000000 possibilities of arranging 52 cards

  3. Miroslav Rác

    What is most likely shuffle for getting the same order of cards as you get after your last shuffling? The very next! https://freethoughtblogs.com/singham/2012/02/06/when-is-lightning-most-likely-to-strike-again/

  4. Balendula

    This video talks about how many possible arrangements of a 52-deck card from beginning to end there are. However, if you want to talk about the PROBABILITY that 2 deck arrangements will be the same from one shuffle to the next, or how many shuffles it will take before you get the same deck arrangement, that is a different story. It's not going to be 1 in 8.0658175e+67 or whatever.

  5. YASSINE PIECE

    amaizang pfffff ohhhhh i watch video 13 time hhh very nice
    روعة و الله شكر كبير للمترجمين ساعة و انا حااااير في فيديو

  6. sanoban

    But if there are so many deck of cards in the world, that would reduce greatly the time needed to get all combinations, no? since there are so many at the same time. Also, you need to realistically remove all possibilities where all suites are together since it would never happen. The number probably would still stay super high, but it might be reduced a bit

  7. We 10 Pandas

    52*51*50*49*48*47*46*45*44*43*42*41*40*39*38*37*36*35*34*33*32*31*30*29*28*27*26*25*24*23*22*21*20*19*18*17*16*15*14*13*12*11*10*9*8*7*6*5*4*3*2=?

    I skipped 1 because anything multiplied by 1 is the same.

  8. Thales pro999

    I play a game called queenie it’s a card game, if u don’t know it it doesnt matter u just need to know getting queens are good, I dealt 5 piles and one had all queens can someone work out the odds of that pls

  9. Hussain

    BUT, WHAT IF BY CHANCE AGAIN, BY CHANCE 2 WELL SHUFFLED DECKS OF CARDS BECOME IDENTICAL. THE ARGUMENT BEHIND IT IS THAT IN THEORY THERE ARE 52! WAYS OF ARRANGING THE DECK OF CARDS BUT FOR ANY 2 DECKS TO BE SIMILAR THEY NEED TO MATCH ONLY ONE COMBINATION. THOUGH IT IS VERY VERY RARE BUT IT IS POSSIBLE!!! THE PROBABILITY OF IT IS HIGHLY LOW BUT YES IT CAN HAPPEN!!
    AGREE!???

  10. Alec Damm

    its not so much that the order you just shuffled has never happened before, theres no way to know that. It is saying there is more possible combinations of deck order than the number of times a deck has been shuffled. by quite a lot

  11. Michael Anderson

    So this leads to another question. What is the PROBABILITY of shuffling an identical deck of cards to one that already exists, and how would you calculate that? Example: you ha a new deck of cards. What is the PROBABILITY that you could shuffle that new deck so that when you are done the cards are back in the original order?

  12. Johannes Bluemink

    Remember the story of the peasant who did a good job for the Emperor of Persia and asked as reward one grain of rice on the 1st square of a Chess-board, 2 on the 2nd, and so forth? I calculated the result by writing it out on a gigantic piece of paper (actually taping 8 A-4's together; that was around 1965). Written out as 2 to the 63rd power, it is a huge number, but these days one can get the answer from Google: 9.223372e+18.

  13. David Rogers

    This doesn't makes sense to me. If you have a deck of 52 cards, and each of those cards can be in one of 52 possible positions. Then it becomes a matter of 52×52 which equals 625. So how do you get 8 to the 67th power?

  14. Bassel Younes

    That not quite right
    Because if pick ace 2 and 3
    It is the same as ace 3 2
    Don't count permutations count combinations which is 52!/13!(52-13)!

  15. kevin a

    Shuffling once per second since the big bang would not even make a dent in the number of possibilities… multiply the age of the universe by 50 trillion and you still have not even made a dent lol

  16. D Kahn

    But, there are also so many combinations that will never be the final outcome from a random shuffle, so you can bring that original number way down….. you are never going to randomly shuffle, 1-2-3-4-5-6-7-8-9 etc… etc…. or 1-3-5-7-9….. or1-4-7-10…. and there are many more…. counting something as a possible computerized combination and actually shuffling them that way are totally different realities.

  17. D Kahn

    This is what makes life so amazing….. think of how many combinations we are made of and then ask what the odds are that we even got to where we are.

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