# of unique hands, not considering suit


The Wikipedia page on cribbage hand says there are 14715 unique cribbage hands not including suit (I assume this includes the cut card, otherwise it's way less)

I wrote a program to enumerate all combinations and list them by type, with a total of 23,647:

The top/notop distinction means the largest run includes the cut card, e.g. a 4K notop is [AAAA, top card 2], while 4K top is [AAA2, top card A]. Hope this makes sense.

Here are the breakdowns by type:
4K no top 156
4K top 156
Full House top 156
FH no top 156
3 of a kind notop 1716
3 of a kind top 858
2P notop 858
2P top 1716
1P notop 8580
1P top 2860
High card 6435

Can anyone match these and verify? The numbers make intuitive sense - for example, there are 13*12*11 2 pair possibilities = 1716. However, when the cut card is not included in a pair, then AABBC and BBAAC are equivalent, so it's 858 unique pairs.

High card = (13! / (8! * 5!)) = 1287. However, any of the 5 can be the top card => 6435 unique.

Also, if the top card is considered just another card, I calculated 6,175 5-card unique combinations. 13*12 4 of a kind, 156 FH, 13*12*11/2! = 858 3 of a kind, 858 2 pair, 13*12*11*10 / 3! = 2860 1 pair, 13*12*11*10*9 / 5! = 1287 high card. Anyone else match this?

Dave Hamilton


Actually Dave,

None of it makes sense. Not even a little teenie weenie little bit. First off, we are talking cribbage here, not poker. There are no Full Houses. There are actually over 12,000,000 (million) combinations of 5 cards you can make in cribbage. Have no idea how you thought it was 14,000 unless you completely eliminate suits, but since flushes count in cribbage that is nonsensical. You HAVE to include suits. Just like you would in poker. Nobody understands your 4k notop or 4k top.