Does not Compute! Trouble w/ Combinatorics

[yellowjack]

I've tried to break down the total number of all possible 5-card poker hands into their categories and get a complete count. I'm a bit short on the total count. Here's what I have:

One Pair: C(13,1)*C(4,2)*C(12,3)*[C(4,1)^3] = 1,098,240

Two Pair: C(13,2)*[C(4,2)^2]*C(11,1)*(4,1) = 123,552

3 of a Kind: C(13,1)*C(4,3)*C(12,2)*[C(4,1)^2] = 24,960

Straight: 10*(4^5-4) = 10,200

Flush: C(4,1)*[C(13,5)-10] = 5,108

Full House: C(13,1)*C(4,3)*C(12,1)*C(4,2) = 3,744

4 of a Kind: C(13,1)*C(4,4)*C(12,1)*C(4,1) = 624

Straight Flush: 4*(10-1) = 36

Royal Flush: 4

High Card: C(13,5)(4^5-4) - 10*(4^5-4) = 1,302,540
where (4^5-4) is all combinations of non-flush hands if there are 5 unique ranks, and 10*(4^5-4) is the number of straights made.

Total = 2,569,008
C(52,5) = 2,598,960 [img]/images/graemlins/frown.gif[/img]

If anyone can spot the error and point it out it would be awesome. The part I had the most trouble with was calculating the number of high card hands, so at least part of the error may lie in here.

[jay_shark]

The only mistake you made was computing 3 of a kind .

(13c1*4c3)*(48*44)/2 = 54912

There are 13c1 ranks and 4c3 possible ways to choose a three of a kind for each . Once this is fixed, there are 48 cards to choose from . Once that is fixed, there are 44 cards to choose from .

Now everything adds up properly .

[yellowjack]

Thanks for pointing it out! Turns out C(13,1)*C(4,3)*C(12,2)*[C(4,1)^2] does work, but I suck at punching numbers in calculators.