For a great training video on poker combinatorics, check out this poker combos video.

'Combinatorics' is a big word for something that isn’t all that difficult to understand. In this article, I will go through the basics of working out hand combinations or 'combos' in poker and give a few examples to help show you why it is useful.

Oh, and as you’ve probably noticed, 'combinatorics', 'hand combinations' and 'combos' refer to the same thing in poker. Don’t get confused if I use them interchangeably, which I probably will.

What is poker combinatorics?

The Texas Hold ’em variant of the poker rules sees players receive two cards, face down, often called ‘the hole’, at the start of the game. The first round of betting occurs at this point. This article will explain everything that I have learned about the subject so far. There are up to 3 total burn cards in a Texas Hold’em poker hand. Before dealing the flop, turn, or river, the dealer is required to. Ultimate Texas Hold’em is played with a standard deck of 52 cards. To begin the hand, the dealer places five cards, known as the community cards, face down on the table. Each player and the dealer are dealt two cards. Players look at their cards.

Poker combinatorics involves working out how many different combinations of a hand exists in a certain situation.

For example:

• How many ways can you be dealt AK?
• How many ways can you be dealt 66?
• How combinations of T9 are there on a flop of T32?
• How many straight draw combinations are there on a flop of AT7?

Using combinatorics, you will be able to quickly work these numbers out and use them to help you make better decisions based on the probability of certain hands showing up.

Poker starting hand combinations basics.

• Any two (e.g. AK or T5) = 16 combinations
• Pairs (e.g. AA or TT) = 6 combinations

If you were take a hand like AK and write down all the possible ways you could be dealt this hand from a deck of cards (e.g. A K, A K, A K etc.), you would find that there are 16 possible combinations.

See all 16 AK hand combinations:

Similarly, if you wrote down all the possible combinations of a pocket pair like JJ (e.g. JJ, JJ, JJ etc.), you would find that there are just 6 possible combinations.

See all 6 JJ pocket pair hand combinations:

So as you can see from these basic starting hand combinations in poker, you’re almost 3 times as likely to be dealt a non-paired hand like AK than a paired hand. That’s pretty interesting in itself, but you can do a lot more than this…

Note: two extra starting hand combinations.

As mentioned above, there are 16 combinations of any two non-paired cards. Therefore, this includes the suited and non-suited combinations.

Here are 2 extra stats that give you the total combinations of any two suited and any two unsuited cards specifically.

• Any two (e.g. AK or 67 suited or unsuited) = 16 combinations
• Any two suited (AKs) = 4 combinations
• Any two unsuited (AKo) = 12 combinations
• Pairs (e.g. AA or TT) = 6 combinations

You won’t use these extra starting hand combinations nearly as much as the first two, but I thought I would include them here for your interest anyway.

It’s easy to work out how there are only 4 suited combinations of any two cards, as there are only 4 suits in the deck. If you then take these 4 suited hands away from the total of 16 'any two' hand combinations (which include both the suited and unsuited hands), you are left with the 12 unsuited hand combinations. Easy.

Fact: There are 1,326 combinations of starting hands in Texas Hold’em in total.

Working out hand combinations using 'known' cards.

How Many Cards Do You Burn In Texas Hold'em

Let’s say we hold KQ on a flop of KT4 (suits do not matter). How many possible combinations of AK and TT are out there that our opponent could hold?

Unpaired hands (e.g. AK).

How to work out the total number of hand combinations for an unpaired hand like AK, JT, or Q3.

Method: Multiply the numbers of available cards for each of the two cards.
Word equation: (1st card available cards) x (2nd card available cards) = total combinations

Example.

If we hold KQ on a KT4 flop, how many possible combinations of AK are there?

There are 4 Aces and 2 Kings (4 minus the 1 on the flop and minus the 1 in our hand) available in the deck.

C = 8, so there are 8 possible combinations of AK if we hold KQ on a flop of KT4.

Paired hands (e.g. TT).

How to work out the total number of hand combinations for an paired hand like AA, JJ, or 44.

Method: Multiply the number of available cards by the number of available cards minus 1, then divide by two.
Word equation: [(available cards) x (available cards - 1)] / 2 = total combinations

Example.

How many combinations of TT are there on a KT4 flop?

Well, on a flop of KT4 here are 3 Tens left in the deck, so…

C = 3, which means there are 3 possible combinations of TT.

Thoughts on working out hand combinations.

Working out the number of possible combinations of unpaired hands is easy enough; just multiply the two numbers of available cards.

Working out the combinations for paired hands looks awkward at first, but it’s not that tricky when you actually try it out. Just find the number of available cards, take 1 away from that number, multiply those two numbers together then half it.

Note: You’ll also notice that this method works for working out the preflop starting hand combinations mentioned earlier on. For example, if you’re working out the number of AK combinations as a starting hand, there are 4 Aces and 4 Kings available, so 4 x 4 = 16 AK combinations.

Why is combinatorics useful?

Because by working out hand combinations, you can find out more useful information about a player’s range.

For example, let’s say that an opponents 3betting range is roughly 2%. This means that they are only ever 3betting AA, KK and AK. That’s a very tight range indeed.

Now, just looking at this range of hands you might think that whenever this player 3bets, they are more likely to have a big pocket pair. After all, both AA and KK are in his range, compared to the single unpaired hand of AK. So without considering combinatorics for this 2% range, you might think that the probability break-up of each hand looks like this:

• AA = 33%
• KK = 33%
• AK = 33%

…with the two big pairs making up the majority of this 2% 3betting range (roughly 66% in total).

However, let’s look at these hands by comparing the total combinations for each hand:

• AA = 6 combinations (21.5%)
• KK = 6 combinations (21.5%)
• AK = 16 combinations (57%)

So out of 28 possible combinations made up from AA, KK and AK, 16 of them come from AK. This means that when our opponent 3bets, the majority of the time he is holding AK and not a big pocket pair.

Now obviously if you’re holding a hand like 75o this is hardly comforting. However, the point is that it’s useful to realise that the probabilities of certain types of hands in a range will vary. Just because a player either has AA or AK, it doesn’t mean that they’re both equally probable holdings - they will actually be holding AK more often than not.

Analogy: If a fruit bowl contains 100 oranges, 1 apple, 1 pear and 1 grape, there is a decent range of fruit (the 'hands'). However, the the fruits are heavily weighted toward oranges, so there is a greater chance of randomly selecting an orange from the bowl than any of the 3 other possible fruits ('AK' in the example above).

This same method applies when you’re trying to work out the probabilities of a range of possible made hands on the flop by looking at the number of hand combinations. For example, if your opponent could have either a straight draw or a set, which of the two is more likely?

Poker combinatorics example hand.

You have 66 on a board of A J 6 8 2. The pot is $12 and you bet $10. Your opponent moves all in for $60, which means you have to call $50 to win a pot of $82.

You are confident that your opponent either has a set or two pair with an Ace (i.e. AJ, A8, A6 or A2). Don’t worry about how you know this or why you’re in this situation, you just are.

According to pot odds, you need to have at least a 38% chance of having the best hand to call. You can now use combinatorics / hand combinations here to help you decide whether or not to call.

Poker combinatorics example hand solution.

First of all, let’s split our opponent’s hands in to hands you beat and hands you don’t beat, working out the number of hand combinations for each.

Adding them all up…

Seeing as you have the best hand 79% of the time (or 79% 'equity') and the pot odds indicate that you only need to have the best hand 38% of the time, it makes it +EV to call.

So whereas you might have initially thought that the number of hands we beat compared to the number of hands we didn’t beat was close to 50/50 (making it likely -EV to call), after looking at the hand combinations we can see that it is actually much closer to 80/20, making calling a profitable play.

Being able to assign a range to your opponent is good, but understanding the different likelihoods of the hands within that range is better.

Poker combinatorics conclusion.

Working out hand combinations in poker is simple:

• Unpaired hands: Multiply the number of available cards. (e.g. AK on an AT2 flop = [3 x 4] = 12 AK combinations).
• Paired hands: Find the number of available cards. Take 1 away from that number, multiply those two numbers together and divide by 2. (e.g. TT on a AT2 flop = [3 x 2] / 2 = 3 TT combinations).

By working out hand combinations you can gain a much better understanding about opponent’s hand ranges. If you only ever deal in ranges and ignore hand combinations, you are missing out on useful information.

It’s unrealistic to think that you’re going to work out all these hand combinations on the fly whilst you’re sat at the table. However, a lot of value comes from simply familiarizing yourself with the varying probabilities of different types of hands for future reference.

For example, after a while you’ll start to realise that straight draws are a lot more common than you think, and that flush draws are far less common than you think. Insights like these will help you when you’re faced with similar decisions in the future.

The next time you’re doing some post session analysis, spend some time thinking about combinatorics and noting down what you find.

Poker combinatorics further reading.

Hand combinations in poker all stem from statistics. So if you’re interested in finding out more about the math side of things, here are a few links that I found helpful:

• Combinations video - Youtube (all the stuff on this channel is awesome)

If you’re more interested in finding out more about combinations in poker only, here are a few interesting reads:

Go back to the awesome Texas Hold'em Strategy.

Comments

• Pineapple - Crazy Pineapple - Crazy Pineapple Hi-Lo

Introduction

Texas Hold'em is a shared card poker game. Each player is dealt two private cards and there are five face up shared (or 'community') cards on the table that can be used by anyone. In the showdown the winner is the player who can make the best five-card poker hand from the seven cards available.

Since the 1990's, Texas Hold'em has become one of the most popular poker games worldwide. Its spread has been helped firstly by a number of well publicised televised tournaments such as the World Series of Poker and secondly by its success as an online game. For many people nowadays, poker has become synonymous with Texas Hold'em.

This page assumes some familiarity with the general rules and terminology of poker. See the poker rules page for an introduction to these, and the poker betting and poker hand ranking pages for further details.

Players and Cards

From two to ten players can take part. In theory more could play, but the game would become unwieldy.

A standard international 52-card pack is used.

The Deal and Betting

Texas Hold'em is usually played with no ante, but with blinds. When there are more than two players, the player to dealer's left places a small blind, and the next player to the left a big blind. The big blind is equal to the minimum bet and the small blind is typically half that amount. When there are only two players (a 'heads-up' game), the dealer places the small blind and the non-dealer the big blind.

The cards are shuffled and cut, the dealer 'burns' one card and then deals the cards one at a time face down until each player has two cards. These are known as the player's hole cards or pocket cards. Players may look at their two hole cards and must not show them to any other player.

How Many Cards Are Dealt In Texas Hold Em

The first betting round is begun by the player to the left of the big blind. The blinds count as bets, so the small blind need only pay the difference between the blinds to call. The big blind player acts last and is allowed to raise, even if the other active players have all called.

After the first betting round the dealer burns one card and deals three cards face up to the table. These three cards are known as the flop. There is then a second betting round begun by the first active player to the left of the dealer seat.

When the second betting round is complete, the dealer burns a card and then deals one card face up to the table. This card is known as the turn or fourth street. There is then a third betting round, again begun by the first active player to the left of the dealer seat.

After the third betting round the dealer burns another card and deals one more card face up - the river or fifth street. There is fourth and final round of betting before the showdown, again begun by the first active player to the left of the dealer seat.

In a fixed-limit or spread-limit game, any betting round that has more than two active players at the start is normally limited to one bet plus three (sometimes four) raises. In the first (pre-flop) betting round the big blind counts for this purpose as the first bet. If a betting round begins with only two active players, there is no limit on the number of raises.

In limit poker, the betting limits are usually doubled before the third betting round. So for example a $10-$20 game would typically have blinds of $5 and $10, bets of $10 in the first two rounds and bets of $20 in the last two.

The Showdown

Active players show their hands in clockwise order, beginning with the player who was the last to bet or raise in the final betting round. If everyone checked in the final betting round, the first active player to the left of the dealer seat is the first to show. See the betting and showdown page for further details.

Each player makes the best possible five-card poker hand from the seven available cards: the player's two hole cards and the five face-up table cards, which are known as the board. This can be in any combination: both hole cards with three from the board, one hole card with four from the board, or just the board cards without using the hole cards at all, which is known as playing the board.

Note that:

• For a hand to be considered for winning the pot, the player must show both hole cards, even if only one or neither of them is used to make the best hand.
• The cards speak for themselves: the best hand is entitled to win the pot if shown, even if the owner does not realise that it is the best hand.
• As always, poker hands consist of just five cards. The two unused cards have no bearing on whether one hand is better than another.

Example. On the board is A-Q-Q-9-6. Player A has K-9; player B has 9-6; player C has 9-3. This is a three-way tie, since all three players have Q-Q-9-9-A as their best hand. The pot is split equally between A, B and C.

Strategy

Because of its great popularity as a tournament and online game, much has been written about the strategy of Texas Hold'em.

The Poker Strategy page provides a list of sites with useful Texas Hold'em Strategy articles.

The Poker Books page reviews a couple of good books about how to play Texas Hold'em.

Variations

The game described above is Texas Hold'em as played formally in public card rooms. When it is played at home, the same procedure can be used, but in some informal games the burning of cards is omitted, and the five community cards are dealt face down to the table at the start. Three of these cards are turned face up after the first betting round, one after the second and one after the third. The result is similar to the formal game, but there is an increased risk that a player might, by accident or by cheating, learn the identity of some of the table cards before they are officially exposed.

The practice of dealing the community cards to the table at the start and turning them face up later is commonly found in community card poker games with more complex layouts, such as Iron Cross and Tic-Tac-Toe. Several of these are listed on the Poker Variants page.

Pineapple

This is played the same way as Texas Hold'em, except that each player is initially dealt three hole cards and must discard one of them before the first round of betting.

Crazy Pineapple

As in Pineapple, each player is dealt three cards, but in Crazy Pineapple they are kept until after the second betting round. Each active player must discard one card immediately before the fourth board card (the turn) is dealt.

Crazy Pineapple Hi-Lo 8 or Better

Crazy Pineapple is often played in such a way that the pot is shared between the players with the highest and lowest hands. The deal and betting are exactly as in Crazy Pineapple. At the showdown, as well as making their best hand, each active player makes the lowest possible hand from their seven cards. When making a low hand, the five cards must all be of different ranks, the aces count as low, straights and flushes do not count, and no card can be higher than an 8. So the best low hand is A-2-3-4-5 and the worst is 8-7-6-5-4. Cards are compared from the top down, so 7-5-4-3-2 is better than 7-6-3-2-A because the 5 is lower than the 6. Players can use a different selection of cards for their high hand and their low hand.

The pot is split equally between the highest hand and the lowest hand. If none of the active players qualifies for low (which will automatically be the case if there are three cards higher than 8 on the board), then the player with the highest hand wins the whole pot.

Irish

This Texas Hold'em variant is similar to Crazy Pineapple, but each player is initially dealt four hole cards. These can all be kept until the end of the second round of betting. Before the fourth board card (the turn) is dealt, each active player must discard two of their hole cards, keeping just two for the turn and river.