# [seqfan] Distinct poker hands

David Scambler dscambler at bmm.com
Tue Oct 5 04:46:28 CEST 2010

```In poker (usually) the suite does not matter except insofar as there are cards with matching suites. For a normal deck of 13 ranks and 4 suites there are C(52, 5) 5-card hands, but when suite symmetries are removed there are only 134459 distinct hands.

Keeping 4 suites and 5-card hands, varying the number n of ranks per suite generates the following number of distinct hands:

S4(n) =  0, 0, 6, 57, 272,901,2376,5362, 10808,19998,34602,56727,88968,134459,196924, 280728,390928,533324,714510,941925,1223904,...

Varying the number of suites and 5-card hands,

1 suite: S1(n) = A000389 = C(n,5)
2 suites: S2(n) = A053132(n+2), n>=3 otherwise zero = C(2n-4,5)
3 suites: S3(n) = 0,0,2,27,152,551,1536,3598,7448,14058,24702,40997,64944,98969,145964, 209328,293008,401540,540090,714495,931304,...
5 suites: S5(n) = 0, 1,12,78,328,1027,2628,5824,11600,21285,36604,59730, 93336,140647,205492,292356,406432,...

Adding a joker to the normal deck

Sj1(n) = 0,1,15,99,405,1231,3072,6671,13070,23661,40237,65043, 100827,150891,219142,310143,...

Adding two indistinguishable jokers to the normal deck:

Sj2(n) = 0,2,21,119,453,1326,3238,6937,13470,24234,41027,66099, 102203,152646,221340,312853,...

Cheers
dave

```