# 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.

## 100 thoughts on “How many ways can you arrange a deck of cards? – Yannay Khaikin”

1. Branana9 - SuperBran says:

Factoriel

2. Marci Reich says:

800 vigintillion possibilitys

3. Titanic says:

so thats what the n! is

4. basiliszag says:

I love the dramatic tone in the explanation of possible permutations!

5. OMG OMG says:

That's 80,658,175,170,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 ways to arrange a card!

6. Zurtron D says:

U did ur math wrong. 7x9x4x5x8x2x7x1=1 regardless the numbers before it it will be one because of the "zero" property of multiplacation

7. Krystof Bristol says:

God coming from Numberphile right to this is like going back to preschool.

8. Gradient Soln-En says:

In about 806,581,751,709,438,785,716,606,368,564,037,669,752,895,054,408,832,778,240,000,000,000 ways

9. Bluestone Randomness says:

What about UNO? 70! = 1.197857166969891796072783721689 * 10^100
(Assuming that each cards are counted separately e.g. there are 4 wild cards)

10. Supertrain12 says:

Easy.
52!.

11. Amathyst t says:

How do find how many combinations for four people to sot in 5 chairs and it can be repeated?

12. Louis Cyphre says:

EXAMPLE : Shuffle cards then take a step , repeat , when you have walked around the planet shuffling the cards every step , take a teaspoon out of the Atlantic ocean , then start walking again shuffling every step, REPEAT this process until every drop of water has been teaspooned out of the atlantic ocean , then start walking again , shuffling the cards every step , but this time once you get around the planet , you put a teaspoon full of water BACK into the Atlantic Ocean , REPEAT this until the Atlantic Ocean is FULL again , then place an A4 piece of paper flat on the ground.
REPEAT this process of walking, emptying, walking , filling , A4 paper , , UNTIL that pile of A4 paper reaches the small dwarf planet named PLUTO. and that is 52 factorial .

13. Steve2018 Gaming says:

If there's a possibility there's a possibility.

14. Pro Meck says:

Just WOW!
Amazing video!!!

15. Hah2game says:

So for the how many arrangements are there for a deck of cards literally all you have to do is the total cards multiplied by the amount of cards in the deck

16. Brandon Baranak says:

Just like the dead mans hand.

17. vũ hạ says:

Giai thừa chứ có gì đâu

18. EnepticX says:

80658175170943878571660636856403766975289505440883277824000000000000 ways

19. Marilor Bastille says:

Hungry for an answer? Table it… like Pi… mmmmmmm PIE

20. Ricez says:

Stand on the equator. Take 1 step every billion years. Once you go all the way around the earth, take a drop of water out of the ocean. Once the ocean is dry, place 1 sheet of paper. Repeat the process. When the sheets of paper reach the sun, that's the same amount of seconds as if you just shuffled the deck every second and finally got the same order once.

21. Orion D. Hunter says:

There are 8452693550620999680 possible arrangements
8.07*10^67=80700000000000000000000000000000000000000000000000000000000000000000

22. juicyguyzer says:

Yannay or Laurel?

23. Malachi Davey says:

I’m a magician, My friend new that so she told me about this video and I think I have an idea for a trick where the performer and a spectator shuffle a deck of cards and the orders are the EXACT SAME!!

24. TheAmazing Sidney says:

Take 52 x 52
Which means
52 cards which can be in any 52 spots

25. Abhi Gamer was taken says:

Wat

26. Nabil Maruf says:

Even all of this I can't win

27. Ace Spade says:

That’s a horrible magnifying glass 2:00

28. J Warthen says:

Or another way to see it is if a deck was shuffled once per second and you had a powerball ticket for a drawing that was once per second, you would win the lottery 4X10*59 times before a shuffle was repeated.

29. Sergej Mit says:

Imagine if you did the same thing with UNO.

30. Antonio Castellanos says:

52 factorial it's huuuge http://bit.ly/2PbFZnK

31. Saayan Biswas says:

One doesn't learn Permutation and combination in just a single day

32. Luke Hayes says:

3:10

33. Kalina Ducharme says:

My caculator does'nt have a n! Or factorial symbol on it…

34. Abandoned says:

Action 52

35. Abandoned says:

4503599627370496, right (I didn't watch the video)

edit: no

36. Abandoned says:

the number is 80658175170943878571660636856403766975289505440883277824000000000000,

btw factorial of 1000 is 402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

37. Limbo Cat says:

“How many ways can you arrange a deck of cards?”
Me: As many as you want to be.

38. Limbo Cat says:

52! Dang that’s a lot of multiplication XD

39. Transendium says:

Oh…the good news is there's more to come, because decimals and real numbers are a thing

40. killerninja360 says:

The answer is 52!

41. dog vader says:

i know on how to beat the system…

just switch the top two cards and keep switching them.

42. Jimmel Matienzo says:

How about uno card deck?
112?

43. Evelia Rodriguez says:

Was I the only one that knew this because of the Reddit post

44. Büşra Akyüz says:

Vay be

45. artemetra says:

52!=80658175170943878571660636856403766975289505440883277824000000000000

46. Wub Monsters says:

What if you shuffle the card with a friend and decide to do it the same?

47. Bloon Crusher says:

about 8 unvigintillion.

48. Михаил Григоров says:

2,19547091E+72 , because the deck has 54 cards

49. Ornithocowian King says:

Weeooh weeooh weeooh! Planetary model of the atom alert! Planetary model of the atom alert!

50. Statiscube says:

Who wants to buy a car? Only £9!

51. mds525700 says:

I deal poker in a casino and I love sharing the factorials of a poker deck when they start crying about being card dead.

52. OofingOofs says:

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

53. Miroslav Rác says:

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/

54. Balendula says:

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.

55. V E G E T A L says:

VSauce was bere…

56. YASSINE PIECE says:

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

57. sanoban says:

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

58. Farzin Karimi says:

Spoilers!
52! Is so so so so so big!

59. Kyle Nguyen says:

What's an intejer

60. Samuele Terzo says:

If we have 3 people are 6 the combinations?

61. Aayushh Dutta says:

52!

62. Aadam Mohammed says:

1000 th comment

63. We 10 Pandas says:

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.

64. Hamid H Shaikh says:

You are an amazing teacher. Thank you.

65. JJ Supreme says:

Me: Holds Pack of cards
Also me: THIS POWER!!!

66. whilly woo says:

Songs

67. whilly woo says:

Spiritual

68. Javier Lopez says:

i am pretty sure that those types of cards dont have 52 types

69. I have Ligma says:

Answer : 52 quintillion

70. Amazing Havlect / ABFDie says:

So 2 be like 2!

71. Tobblesmash says:

3:18 love that animation

72. Matthew Ramos says:

80658175170000000000000000000000000000000000000000000000000000000000 is 52!

73. Ersatz Gemini says:

https://www.youtube.com/channel/UCT0ICZfNieiJg0cmA3qKFDA
Ersatz Gemini is my channel name please check it and subscribe

74. Dark Prince says:

Mind blown

75. Lukine Gluposti says:

8.068e67

76. christian Saldaña says:

How to get the possibilities: Get the number's factorial.

77. Farid Ahmed Deputy says:

I learned factorials basics from here
Tnkh u😎

78. rtyrty12 says:

1 x 2 x 3 …. x 52 = The massive number of arrangements a deck of cards can be made.

79. Thales pro999 says:

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

80. Hussain P says:

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!???

81. Pranjal Tiwari says:

Hi ayush

82. Charles Long says:

I know I have received the exact same bridge hand several times, and I [email protected] played bridge for 52 years.

83. Alec Damm says:

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

84. spoi crab says:

Amount of cards allowed in hand x amount of total cards

85. Marshall Kimber says:

But what if you arrange it in a deck specifically an order that you remembered in that order again
🤔🤔🤔

86. Name LastName says:

And Go Fish has suddenly become that much more troubling

87. Discovaria says:

Many many ways

88. 2013064 tskp says:

No time to learn
Factory must grow

89. Sunep Imchen says:

Wow interesting

90. Michael Anderson says:

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?

91. cubeing says:

52!=80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000

92. Jade White says:

No wonder getting a royal flush is so hard

93. Soviet Thundurus says:

Fun fact: If you do a faro shuffle eight times, the shuffled deck will be restored to its original order.

94. Wrath Black says:

4 x 3 = 12 x 2 = 24.
52 x 51 = 2652 x 2 = 5304.

95. The Underrated Guy says:

The video of Vsauce related to this video is also very informative.

96. Tan CHUANG KI says:

I know permutation but never thought 52! is a huge number !!!

97. Hamdaan Chalky says:

Are you challenging my extreme luck?!?!!?!!

98. yelshAj • 20 years ago says:

80.6 unvigintillion arrangements for a deck of 52 cards.

99. Jesse Jenner says:

I'm looking at a deck of cards right now, having a hard time grasping that number. wild.

100. Jordan ZAFIROPOULOS says:

Okay now this is epic!