More ways to arrange a deck of cards than atoms on Earth

[dlevere]

There are roughly 80,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000 unique ways to order 52 playing cards. “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.”

By Mark Frauenfelder

[xirtamehtsitahw]

Here's another shocker: it only takes SEVEN shuffles to create a completely random arrangement of cards in a standard, 52-card deck. The only way so few shuffles can do so is to use the two-hand zipper-type shuffle (the cards rapidly fit together like the teeth of a zipper). Simply "shaking" the cards a few at a time, thus rolling them over and over each other, would take around 10 to 15 MINUTES to achieve the same level of randomness.

Also, there are 2,598,960 unique poker hands (assuming one is playing 5-card poker). Since there are only four possible Royal Flush hands, you have less than a 1 in 600,000 chance of being dealt one. It's a hand so rare, that if you receive it, you may be accused of cheating!

One last thing: that 80 with 66 zeros after it has an official name (though very few would know it, let alone know how to spell it):

80 unvigintillion. The name is much shorter than the number itself, but it is a name that is extremely rarely heard outside of high-level mathematical circles...or possibly a MENSA meeting.

[bungholio]

Doesn't a factorial of 52 just give you the exact amount of possibilities?

[dlevere]

No clue on that one. Check this out:

https://www.khanacademy.org/math/pre...ations--part-2

[bungholio]

I haven't done math in forever, but factorial or whatever it was gives you the exact amount or permutations. It was 52! or !52.
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 = some really big number

[xirtamehtsitahw]

Actually, bungholio's right--52! (read "52 factorial") is the exact number of ways a deck of 52 cards can be arranged (as there are no duplicates in a single deck). Duplicates slightly complicate the process, and require the use of one of the following formulas: n!/(n-r)! or n!/(n-r!)r!. The first one is for permutations and the second one is for combinations. A permutation does not take into consideration the possibility of identical arrangements of items, merely the number of items, taken a group at a time. A combination DOES take such identical arrangements into account, and eliminates all but the first such arrangement.

The highest factorial allowed on a 99-digit scientific calculator (allows for two-digit scientific-notation exponents) is 69!. This number is around 10 to the 97th power (give or take a few--can't remember exactly). On a calculator capable of handling up to three-digit scientific-notation exponents, the highest factorial possible is well over 400 (can't remember the exact number).