how many possible hands in a deck of cards

[Poker21]

Can anyone tell me how many possible 5 card hands are in a 52 card deck?

[jpym84]

The number of possible hands is the number of possible combinations of 5 cards given a 52 card deck, where each unique ordering of the cards isn't counted (i.e. Ts-Js-Qs-Ks-As is the same hand as As-Js-Ts-Qs-Ks). So the number of hands is:
52!/((52-5)! * 5!)
=52! / (47! * 5!)
=(52*51*50*49*48) / (5*4*3*2*1)
=2598960

[scottjack]

There's 2,598,960 possible 5 card hands out of a deck of 52 cards.

The formula is often abbreviated by nCr (where n represents the number of things you're counting, C stands for 'Combination', and r is how many of the n things you want to count at a time) In our case, n is 52, r is 5.

The formula goes like this: nCr = n!/[r!*(n-r)!]

The exclamation point means you take the number and multiply it by itself and each number between itself and 1.

(5! would be 5x4x3x2x1, 52! would be 52x51x50...all the way to 1)

So the calculation is nCr = 52!/[5!x47!]

The formula is in most any decent spreadsheet. To see the details involving the math behind the formula, look up 'combinations' in most any math site.

Hope this is helpful!

[well]

[ QUOTE ]
The exclamation point means you take the number and multiply it by itself and each number between itself and 1.

[/ QUOTE ]

You won't multiply the number by itself!
This would make 5! = 5*5*4*3*2, which is in fact 4 times too much!

I understand that it is not that you do not understand...

[scottjack]

you're correct, of course..i only meant to be certain that the original number was included in the calculation.
Thanks for the correction.
/images/graemlins/blush.gif

[DeadRed]

The number 2598960 is well-known, this has always (and properly) been explained that it counts hands that are the same poker-wise as being different. E.g. the 4 royal flushes as 4 different hands. My initial computation on counting the different ranking poker hands has yielded 8154 different ranks of hands, from a royal flush (4 possible ways to make) to 7-5-4-3-2 (1020 possible ways to make without being flush). I am not sure of the accuracy of this number yet, because my double-checking is not coming out right. I will update this when all the numbers jibe.

[DeadRed]

i am updating the number to 7462 hands. My double-check says I am missing 896 out of the ~2.6 million card combinations, so I think I am close!

[BugsBunny]

Lets see if I understand what you're trying to figure here:

Straight flush, including royal = 10 (A, K, Q, J, T, 9, 8, 7, 6, 5 high)

4 of a kind = 13 * 12 = 156 (4 of each rank * 12 different kickers)

Full house = 13 * 12 = 156 (3 of each rank * 12 different pairs)

Flush = C(13,5) - 10 straight flushes = 1287 - 10 = 1277

straight = 10 (see straight flush)

trips = 13 * C(12,2) = 13 * 66 = 858

2 pairs = 11(12+11+10+9+8+7+6+5+4+3+2+1) = 858 (you have 11 different kickers for each 2 pair combination. AA has 12 additional pairs, KK has 11, QQ has 10 down to 33 which has 1)

pairs = 13 * C(12,3) = 13 * 220 = 2860

High card = C(13,5) - 10 straights = 1287 - 10 = 1277



Total = 10 + 156 + 156 + 1277 + 10 + 858 + 858 + 2860 + 1277 = 7462

[DeadRed]

That is exactly how I got my number, but my double-check is not coming out right. To make reading easier, let me make this distinction: A "hand" is a card ranking, while a "combination" is a collection of 5 cards. In my terminology the top 5 hearts and the type 5 spades are the same hand, but different combinations. So there are 7462 hands and 2598960 combinations. To verify the number of hands, I am trying to count the number of combinations that make each hand to get a total of 2598960.

[ QUOTE ]
Lets see if I understand what you're trying to figure here:

Straight flush, including royal = 10 (A, K, Q, J, T, 9, 8, 7, 6, 5 high)



[/ QUOTE ]

Each straight flush can be made in 4 suits so there are 10 X 4 = 40 combinations that make straight flushes.

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4 of a kind = 13 * 12 = 156 (4 of each rank * 12 different kickers)


[/ QUOTE ]

There is only 1 combination of quads, but the kicker can be of any of the suits, so 156 X 4 = 624 combinations.

[ QUOTE ]

Full house = 13 * 12 = 156 (3 of each rank * 12 different pairs)


[/ QUOTE ]

Each set can be made 4 ways and each pair 6 different ways, so 156 X 24 = 3744 combinations.

[ QUOTE ]

Flush = C(13,5) - 10 straight flushes = 1287 - 10 = 1277


[/ QUOTE ]

One flush in each suit. 1277 X 4 = 5148 flush combinations.

[ QUOTE ]

straight = 10 (see straight flush)


[/ QUOTE ]

There are 4^5 ways to suit the 10 cards, but 4 of these make a straight flush, so 1020 combinations for each straight. 10 X 1020 = 10200

[ QUOTE ]

trips = 13 * C(12,2) = 13 * 66 = 858


[/ QUOTE ]

Each set made in 4 ways, each kicker can be of 4 suits, so 64 combination for each hand. 858 X 64 = 54912

[ QUOTE ]

2 pairs = 11(12+11+10+9+8+7+6+5+4+3+2+1) = 858 (you have 11 different kickers for each 2 pair combination. AA has 12 additional pairs, KK has 11, QQ has 10 down to 33 which has 1)


[/ QUOTE ]

Interesting that you didn't use C(13,2) * 11 kickers to get this number, but it shows an interesting fact: 1+2+3+...+n = C ( n+1 , 2 ). Anyway, each of the two pair has 6 combinations and the kicker 4 suits, so 144 combination per hand. 858 X 144 = 123552

[ QUOTE ]

pairs = 13 * C(12,3) = 13 * 220 = 2860


[/ QUOTE ]

6 ways to make each pair, 4 suits for each of the 3 kickers, or 384 combos/hand. 2860 X 384 = 1098240

[ QUOTE ]

High card = C(13,5) - 10 straights = 1287 - 10 = 1277


[/ QUOTE ]

1020 for each. (See straights above)

1277 X 1020 = 1302540

40+624+3744+...+1302540 = 2599000 which is 40 too many! It's driving me crazy!
/images/graemlins/confused.gif

[well]

[ QUOTE ]

1277 X 4 = 5148


[/ QUOTE ]

[DeadRed]

THANK YOU!! I was so tired of going through those numbers that I didn't want to run through them all again. I either can't type, read, or write. Can't figure out which. It changes periodically. /images/graemlins/crazy.gif

[donkeyradish]

I guess in stud the number of rankings is much fewer

Eg. If you have trips the kickers are irrelevant because only one person at a time can ever have 333.

[DeadRed]

No, this number is game-independent. As long as you are playing 5 cards to make a hand (None of that crazy "I have 3 pair" business) then this count is accurate. There are 7462 different poker hands as listed in the posts above.