How Many Ways To Arrange a Deck of Cards

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John Tudor
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How Many Ways To Arrange a Deck of Cards

Postby John Tudor » February 20th, 2017, 10:36 pm

There Are More Ways To Arrange a Deck of Cards Than Atoms on Earth

https://www.youtube.com/watch?v=uNS1QvDzCVw

The TED talk claims there are 80,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 (67 zeros) ways to arrange a deck of 52 cards. And as Khaikin explains, "Any time you pick up a well shuffled deck, you are almost certainly holding an arrangement of cards that has never before existed and might not exist again."
"Ars longa, vita brevis." (Life is short, the art so long to learn) -Hippocrates

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Bob Cunningham
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Re: How Many Ways To Arrange a Deck of Cards

Postby Bob Cunningham » February 21st, 2017, 9:46 am


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Re: How Many Ways To Arrange a Deck of Cards

Postby erdnasephile » February 21st, 2017, 10:12 am

"Any time you pick up a well shuffled deck, you are almost certainly holding an arrangement of cards that has never before existed and might not exist again"

I wonder if that is really true. Given that USPCC starts every deck in a certain order, wouldn't certain deck arrangements tend to be more common within, say, the first 7 to 10 shuffles? Also, wouldn't certain arrangements (e.g., new card order with only 2 cards out of place) occur less commonly out in the wild?

(I'm not a mathematician, so I'd love it if someone would enlighten me here)

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Re: How Many Ways To Arrange a Deck of Cards

Postby Bill Mullins » February 21st, 2017, 10:53 am

erdnasephile wrote:"Any time you pick up a well shuffled deck, you are almost certainly holding an arrangement of cards that has never before existed and might not exist again"

I wonder if that is really true. Given that USPCC starts every deck in a certain order, wouldn't certain deck arrangements tend to be more common within, say, the first 7 to 10 shuffles? Also, wouldn't certain arrangements (e.g., new card order with only 2 cards out of place) occur less commonly out in the wild?

(I'm not a mathematician, so I'd love it if someone would enlighten me here)


I think these outcomes are ruled out by "well shuffled deck". (and from Diaconis's work, we know that it takes about seven good riffle shuffles to get to that "well shuffled" state.)

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Re: How Many Ways To Arrange a Deck of Cards

Postby brianarudolph » February 21st, 2017, 11:18 am


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Re: How Many Ways To Arrange a Deck of Cards

Postby MagicbyAlfred » February 21st, 2017, 1:50 pm

Since we are unlikely to hear from Percy Diaconis on this, I will throw in my two cents based on my relatively modest training in statistics during college. I could well be mistaken, but I believe that to determine the number of possible combinations in a deck of 52 cards, one would need to calculate 52 to the 52d power. 52 to the 2d power (or 52 squared) = 2704, 52 to the 3d power is 52 x 2704 = 140,608. 52 to the 4th power is 52 x 140,608 = 7,311,616. 52 to the 5th power is 52 x 7,311,616 = 395,804,032. 52 to the 6th power = 20,581,809,664, So we are already over twenty billion and counting!

Imagine continuing that process 45 more times!

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Re: How Many Ways To Arrange a Deck of Cards

Postby MagicbyAlfred » February 21st, 2017, 1:51 pm

MagicbyAlfred wrote:Since we are unlikely to hear from Percy Diaconis on this, I will throw in my two cents based on my relatively modest training in statistics during college. I could well be mistaken, but I believe that to determine the number of possible combinations in a deck of 52 cards, one would need to calculate 52 to the 52d power. 52 to the 1st power = 52. 52 to the 2d power (or 52 squared) = 2704, 52 to the 3d power (or 52 cubed) is 52 x 2704 = 140,608. 52 to the 4th power is 52 x 140,608 = 7,311,616. 52 to the 5th power is 52 x 7,311,616 = 395,804,032. 52 to the 6th power = 20,581,809,664, So we are already over twenty billion and counting!

Imagine continuing that process 46 more times!

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Re: How Many Ways To Arrange a Deck of Cards

Postby Bill Mullins » February 21st, 2017, 2:47 pm

MagicbyAlfred wrote:Since we are unlikely to hear from Percy Diaconis on this, I will throw in my two cents based on my relatively modest training in statistics during college. I could well be mistaken, but I believe that to determine the number of possible combinations in a deck of 52 cards, one would need to calculate 52 to the 52d power. 52 to the 2d power (or 52 squared) = 2704, 52 to the 3d power is 52 x 2704 = 140,608. 52 to the 4th power is 52 x 140,608 = 7,311,616. 52 to the 5th power is 52 x 7,311,616 = 395,804,032. 52 to the 6th power = 20,581,809,664, So we are already over twenty billion and counting!

Imagine continuing that process 45 more times!


It's Persi, not Percy.

Not 52^52, but 52! (factorial).

The first place in the arrangement can be occupied by any of 52 cards. The second can be occupied by any of the 51 remaining. The third by any of the 50 remaining . . . . and so on and so on. 52 x 51 x 50 x 49 . . . . 3 x 2 x 1.

52^52 ≈ 1.7 x 10^89

52! ≈ 8 x 10^67

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Re: How Many Ways To Arrange a Deck of Cards

Postby MagicbyAlfred » February 21st, 2017, 3:24 pm

I think I will just stick to being an entertainer, being not as smart as others on here...

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Re: How Many Ways To Arrange a Deck of Cards

Postby John Tudor » February 21st, 2017, 4:15 pm

Alfred, me neither!
"Ars longa, vita brevis." (Life is short, the art so long to learn) -Hippocrates



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Re: How Many Ways To Arrange a Deck of Cards

Postby MagicbyAlfred » February 21st, 2017, 5:35 pm

John Tudor wrote:Alfred, me neither!


Thanks John, you are kind!

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Re: How Many Ways To Arrange a Deck of Cards

Postby performer » February 21st, 2017, 7:28 pm

Oddly enough that is how the Tarot works. If ten cards are chosen by the client at random the odds are astronomical that those specific cards wold be withdrawn and laid out in a particular combination. It makes each reading unique and you fit the cards to the client when you interpret them. I believe it works on heightened intuition and the cards come up by pure chance but another school of thought relies on Jung's theory of synchronicity. I don't necessarily go along with that even if I knew what it was! Something to do with time and space and the particular cards can only be chosen at that particular moment. I shall have to look it up.


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