Interesting video for card workers...seems like good patter for certain tricks. The sheer combination of arranging 52 cards in various ways is astronomical.

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

Cool vid! Thanks for sharing.

At least 100

52 factorial

I'm pretty sure it's more than that.

52 factorial is a lot, 80000000000000000000000000000000000000000000000000000000000000000000 possibilities!!!

Ask Tom Ransom.

Wouldn't it be 52x51x50x49x48x47 --------------------------------------------------x5x4x3x2x1?

You're right Harry. That's what 52! (fifty two factorial) means viz. 52x51x50.... 5x4x3x2x1. The approximate value is 8 x 10^67 (ten to the power 67) or 8 followed by 67 zeros. On the video the guy says that that is more than the number of atoms in the earth which is true but fails to convey how much more.

A fair estimate of the number of atoms in the earth is 9x10^49. I have seen various values but they don't differ enough to be important. 8x 10^67 is 10^18 larger. Thus it would take 10^18 earths (that's a million trillion) to have enough atoms to equal the number of arrangements of a deck of 52 cards. The number of stars in our galaxy is about 200 million. That's 2x10^8. So if every star in our galaxy had an earth, there still wouldn't be enough atoms in all those earths to equal 52! We'd need abou 10^10 galaxies like ours. That's ten billion galaxies with each star having a planet like earth. Then all the atoms of all those earths would be about the number of arrangements of a deck of cards.

So it's fair to say that no well shuffled deck has ever or will ever be in the same order as another one in the entire history of the human race into the future. The probability is so small that it should be considered zero. It would be like if all the atoms of all the earths on all the stars in the ten billion galaxies were mixed together and we picked one and then mixed them again. It's the odds of picking the same one a second time.

Mike

This number is so mind boggling - my brain hurts thinking about it But I like how you try to put it into perspective Mike!

What I Simply LOVE, then, is the fact that when I perform UpThe Ante, even though the spectator has 52! (8.07 x 10 to the power 67) Ways of trying to arrange the deck so as to screw me up, I still am in COMPLETE control! That is mind boggling.....and very, very reassuring at the same time! I must bring this number into the patter.

I second what Captain Smiffy says about Up The Ante. As someone who performs it regularly, I think I'll try to incorporate that number into the presentation. I think it would work well for the right audience.

Some more fun facts when it comes to large numbers...people often lump big numbers together, like "million" or "billion", what's the difference, right? To illustrate the difference, I ask them how many days do they think it takes for a million seconds to tick by. They usually respond "a few days," and they're close. It's about 11 days. Then I ask them how many days for a billion seconds. I get answers ranging from a month to a year, but nobody gets the right answer: 32 years. And a trillion seconds? 32,000 years. IIRC, I first read that in John Allen Paulos' book "Innumeracy: Mathematical Illiteracy and Its Consequences".

In regards to comparing the number of different ways to shuffle a deck to the atoms in the earth, I've seen something similar in regards to the number of possible chess games. I think I read something years ago by Isaac Asimov that if you played a complete game of chess every second, there is not enough time left in the universe to play them all. In other words, given current cosmological understanding (at least when Asimov wrote his article), the universe will reach the end of its "life" (absolute zero throughout, no movement of gas molecules, etc.) and there will still be chess games to be played.

Sorry to have gotten too far afield from the original discussion.

- Bob

Actually all wrong. While 52! is correct for a normal shuffled pack of cards, it doesn't include whether or not the card is face up or face down. Which means(for, for example, a triumph-type shuffling) the actual number is more like 52! x 2^52

Why would you say it's "Actually all wrong.", and then admit that for the conversation taking place, it's exactly correct? No one brought up a triumph routine, and in the context of patter for audiences, they wouldn't know why you'd ever consider cards face up and face down. For most triumph routines, the number of possible mixes doesn't even make sense.

Next time, just say "hey, if you use a triumph routine where you then make then entire deck go back into order, you can use an even bigger number for the probability."

Don't forget the jokers.

Argh, I see that the OP says 52 cards. Oops.

How about in Vegas where they use 6 decks in a shoe. Unfortunately the fact that there are 6 duplicate cards of each type now complicates the calculation. It won't be 312! or (six times fifty two)! That would occur if all 312 cards were different. If the decks had different back designs, then 312! would be the number.

I doubt that calculators can handle 312! Just tried it on my iPhone scientific calculator. It's too big. It'll be WAY bigger than the number of atoms in the entire universe.

100! is 9 x 10^157. The online calculator says that 200! is INFINITY. 312! is WAY WAY bigger than that. The power of 10 will be HUGE. The number of atoms in the entire known universe is between 10^78 and 10^82. Don't forget that 10^88 would be a million (10^6) times bigger than 10^82.

If you shuffle together a red deck and a blue deck each of 52 different cards, there would be 104! different arrangements which is about 10^166. Yikes.

Mike

Hi Mike,

When the six decks are identical, the number of arrangements is simply 312! /((6!)^52). To see this, just notice that 6! is the number of way you can arrange each set of identical cards. In any case, it is still a huge number. To get an idea of how n! grows, you can use Stirling's formula which states that n! grows like sqrt(2pi n)(n/e)^n . For the particular cases here, the number of arrangements for 6 decks with different backs is about 10^644 whereas it is about 10^495 when all the backs are identical. In any case, these are (more than!) astronomical number...

Cheers,

Thanks for the more refined calculations! I forgot about Stirling's formula. I noticed that someone has a C++ program to calculate factorials of large numbers. The on-line calculator gives up after 150!

Mike

The Persi Diaconis book has it at less than 52! and explains the reasoning.