Hello,

I am using Excel and some php code to run out all possible made hand (preflop+postflop) combinations e.g.:

High Card Preflop:
High Card, Three - Kicker 2
High Card, Four - Kicker 2
High Card, Five - Kicker 2

One Pair Preflop:
One Pair, Twos
One Pair, Threes
One Pair, Fours

High Card Postflop:
High Card, Seven - Kicker 5432
High Card, Seven - Kicker 6432
High Card, Seven - Kicker 6532

...

The result is 7220...is this correct?

Preflop: 13C2 + 13 = 91

Postflop:
--------
First let's count the impossible.
For instance 7-5-4-3-2 is impossible because you're making the best hand out of 7 cards and any other card automatically gives you a better hand than that.

Also impossible are:
7-6-4-3-2
8-5-4-3-2
8-6-x-x-x = 4C3 = 4 hands
8-7-x-x-x = 5C2 - 1 = 9 hands (the -1 is the straight which is possible)
...
actually there are a lot to consider, so I'll continue this another time.

And then with the One Pair hands, I think the number of impossible kickers depends on which pair you have.

Not a fun problem heh.

I feel like something like this would have been answered before. Maybe by Brian Alspach.

What is the definition of "made hand" in this context?

There are 7462 five-card hands:

13C5 - 10 = 1277 no-pair hands.
13*12C3 = 2860 one-pair hands.
13C2*11 = 858 two-pair hands.
13*12C2 = 858 trip hands.
the 10 straights we subtracted from the no-pair hands;
1277 flushes (all the no-pair hands, suited);
13*12 = 156 full houses;
13*12 = 156 quads;
10 straight flushes.

How many are possible on the river is a harder question. Given that OP included 75432 etc in his list (and on the river 98754 is the worst possible hand) I think that 7462+91 may be the answer he had in mind.

In this context I looked at:

1) All possible preflop hands
- High Card + Kicker (1)
- One Pair
2) All possbile postflop hands
- High Card + Kicker (4)
- One Pair + Kicker (3)
- Two Pair + Kicker (1)
- Three of a Kind + Kicker (2)
- Straight
- Flush + Kicker (4)
- Full House
- Four of a Kind + Kicker (1)
- Straight Flush
- Royal Flush

Shouldn't "High Card, Seven - Kicker 5432" be the worst possible hand on the River (assuming only 1 player goes to showdown, what of course will never happen)? I am just trying to list all possible hands without considering multiple players in the hand.

Ok. So 2) is essentially the same question as "how many different 5 card poker hands are possible"?

e: If that is correct, and using Alspach (http://people.math.sfu.ca/~alspach/comp18/) as a basis, I get:

10 different straight flushes
156 different four-of-a-kinds
156 different full houses
1277 different flushes
10 different straights
858 different three-of-a-kinds
858 different two-pairs
2860 different pairs
1277 different high-cards

for a total of 7462 different hands

e2: I see that Siegmund already said that

Right now I am getting this:

1) All possible preflop hands
- High Card + Kicker (1) = 78 possible hands
- One Pair = 13 possible hands
2) All possbile postflop hands
- High Card + Kicker (4) = 1277 possible hands
- One Pair + Kicker (3) = 2860 possible hands
- Two Pair + Kicker (1) = 832 possible hands
- Three of a Kind + Kicker (2) = 1716/2 trips or set = 858 possible hands
- Straight = 10 possible hands
- Flush + Kicker (4) = 5108/4 clubs or diamonds or hearts or spades = 1277
- Full House = 78 possible hands
- Four of a Kind + Kicker (1) = 156
- Straight Flush = 36/4 clubs or diamonds or hearts or spades = 9
- Royal Flush = 4/4 clubs or diamonds or hearts or spades = 1

7449 possible hands not considering the suits

12168 possbile hands considering the suits

I have to check Two Pair and Full House

@Siemund
@hautari

Have you considered to discount the Kikcer hands with Two Pair e.g.

Two Pair, Threes and Twos - Kicker 4
Two Pair, Fours and Twos - Kicker 4 (Discount)
Two Pair, Fives and Twos - Kicker 4

Thats a full house, it's not counted as two pair.

You should recount two pairs and full houses, the correct numbers are 858 and 156, respectively

Sorry for the brief response, I had to attend a meeting. More thorough response:

- two pairs: 13*12/2 = 78 kombinations, each of which can have 11 different kickers = 78*11 = 858 kombinations

- full houses: 13*12 = 156 kombinations (AAAKK and KKKAA are different hands etc, thus no division by 2)

No, because the two other cards not considered would have to improve the hand.

Oh, that's true...on the river...on the flop, High Card, Seven - Kicker 5432 would be the worst possible hand...since it is a possible hand, it should be listed.

I found the missing Two Pair and Full House hands in my list, numbers now add up! Thank you to all

Finding the list of possible river hands (best 5 of 7) is an interesting problem, too, and one I spent a little while on this afternoon.

All straights or better are still possible. With trips and worse, we kick out the lowest two cards, but have to make sure those two cards don't make a straight.

I am confident in the general method, but it's awfully easy to miss a few special cases, so don't take these as gospel unless you doublecheck carefully:

For high card, we consider 11C5=462 possibilities for the highest 5 of 7 unpaired cards, but we have to remove A**54 (28 hands), **654 (21 hands, not doublecounting the aces), and AKQJT through 98765 (6 hands) leaving 407 ways to have no pair.

With one pair, we consider each of 13 pairs, and 10C3=120 possible highest three of five unpaired cards.
With 22, we must remove *65 and A*5, as the 4 and 3 will make a straight. That leaves 105 of the 120 22xyz hands.
Similarly for 33, 44, and 55.
With 66, we remove *54, leaving 112 hands.
With 77 through KK, we remove only the single combination 654, leaving 119 hands each. 4x105+2x112+7x119=1477 hands.
With AA we remove *54 leaving 112 hands.

With two pair, we generically have 13C2 two-pair hands with 9 possible highest kickers each. But we must exclude 33226 (the 5-4 would make a straight), 44226, 44336, 55226, 55336, 55446; 66225, 66335, 66445, 66554; AA225, AA335, AA445, and AA554. Eight kickers for 14 hands, nine kickers for the other 64. 688 possible hands.

With trips, we generally have 10C2=45 possible holdings for the two highest of four missing cards, but
For 222, we must exclude 6-5 and A-5 as the 4-3 would make a straight;
similarly for 333-555.
For 666 and AAA, we must exclude 5-4; from 777-KKK, nothing. 575 possible hands.

So, plus or minus special cases I stumbled over, we have
407 no-pairs left out of 1277;
1477 one-pairs left out of 2860;
688 two-pairs left out of 858;
575 trips left out of 858;
and all 10 straights, all 1277 flushes, all 156 full houses and quads, all 10 straight flushes, for 4756 of the 7462 hands still possible on the river.

I was just looking at an ordered list of all possible made hands preflop (not really considered a made hand) and postflop, but looking at the river is interesting as well, since two additional cards will cause some hands to drop out of this list Thank you...