Hi All,
I pulled this quote from another thread, but the information it requests might be of interest to more people than just the 1 who asked.

The 2.59million poker hands what is this? Is this combinations of flops or something? :f_ugly: sounds interesting and I'm curious!
I had mentioned something previously about there being only 2598960 different combinations of cards that create all poker hands. There actually 8.09 x 10^67 different ways to order 52 cards in a deck.

If you dealt 100 hands a second you would not work your way through all of them in the estimated life of the universe.

That does not matter because the order in which the cards appear has no bearing on the value of the hand. Furthermore, for all hands except flushes and straight flushes, the suit is irrelevant. (Let's leave Badugi out of this -- it makes my brain hurt).

The 2598960 is not the number of possible hands -- just the number of different combinations of cards that give poker hands.
In fact there are only 7462 distinct hand values.

For example, there are only 10 possible Straight Flushes -- from A-high down to 5-high.
Since there are 4 suits, there are 4 possible A-high straight flushes and so on.
So 40 combinations all yield the same hand type, 4 of each different rank.

Now let's do 4-of-a-kind.
For AAAA there are 12 possible kickers in 4 possible suits, so there are 48 combinations that give quad Aces.
For quads, the kicker is irrelevant except in community card games like Omahahaha and Hold'em, and all four Aces are on the board. The point is that all 48 combinations give EXACTLY the same hand.
So 13 ranks, 48 combos each = 624 combinations of quads.

You get the idea.
I won't derive the rest, however...
There are 156 distinct Full Houses with 24 combos each for 3,744 combinations
There are 1,277 distinct flushes with 4 combinations each for 5,108 combinations
There are 10 distinct straights, with 1020 combinations each for 10,200 combinations
There are 792 distinct 3-of-a-kind hands with 64 combinations each for 50,688 combinations
There are 858 distinct 2-pair hands, with 144 combinations each for 123,552 combinations.
There are 2860 distinct 1-pair hands, with 384 combinations each for 1,098,240 combinations.

Now we come to the garbage. Well it is garbage unless you're playing a Hi/Lo game, Razz or some other lowball game.
The hands such as AKQJ9 that have no pair, and no flush make up over half of all possible hands.
For each distinct no-pair-no-flush hand there are 1020 combinations that yield that hand.
There are 1,277 of these hands, thus there are 1,302,540 no-pair-no-flush hands.

reference: http://people.math.sfu.ca/~alspach/comp18/

I hope that interests someone.