Combinatorics is a branch of mathematics that deals with objects belonging to a finite set. In poker that finite set is the number of possible starting hand combinations.
Poker is no longer just a game about staring down your opponent and trying to figure out if they have it. Mathematics has become an increasingly important aspect of poker strategy, and the math of 'counting combos' or 'poker combinatorics' is what we're going to break down today.
Poker Combinatorics Explained
Combinatorics is a branch of mathematics that deals with objects belonging to a finite set. In poker that finite set is the number of possible starting hand combinations, and advanced players will use combinatorics to figure out how likely certain hand combinations are in their opponent's range, given the information they have.
Texas Hold'em combinatorics are relatively simple to figure out, given that a hand can only consist of two cards and there is a maximum of four cards of a single value. There are a total of 169 different starting hands in Texas Hold'em with 1,326 distinct combinations of those hands. Read the Wikipedia article about poker probabilities to have a solid foundation.
For each individual starting hand (such as AA or AK), there are multiple ways that the hand can be made. For example, you can make a total of 6 combinations of AA like so:, , , , , and . This is how we get 1,326 unique combinations from 169 different hands.
That may sound like a lot but if you compare it to Pot Limit Omaha where there are 270,725 distinct combinations of hands, it suddenly seems a lot more manageable! The more cards you have in your hand, the more complex poker combinatorics becomes.
Advantages of Using Poker Combinatorics
Using poker combinatorics at the table will give you a great advantage over players who don't as it gives you a mathematically sound method of deducing the likelihood of different hands being in your opponent's range.
Before poker combinatorics was widely used, players would only be able to guess how often a certain hand appears in their opponent's range based on how the hand played out until that point. Hand reading like this is still used, but in conjunction with poker combinatorics you can figure out the number of combos your opponent is likely to have.
If you can figure out the number of hand combinations your opponent has, you can calculate the number of bluffs and value bets you think they have and then relate that to the pot odds you're being laid to see if it's profitable to call.
For example, you're on the river in a hand and your opponent has just bet half pot. We know from our poker math knowledge that to be profitable we need to win at least 25% of the time. Using combinatorics we think our opponent is betting 40 combos for value and 20 combos as a bluff. This means if we call we will win 33% of the time. Given that we only need to win 25% of the time, we can make the call knowing it will be profitable.
Accounting the Preflop Hand Combinations
Each type of hand has a different number of total combos which you'll need to remember if you're going to use combinatorics at the table.
Each pocket pair has a total of 6 combinations that can be made from the four suits, like so;, , , , , .
There are 13 different varieties of pocket pairs, meaning that there are 78 unique pocket pair combinations - only around 6% of the total hand combinations.
This means that when you're thinking about your opponent's range, any potential pocket pairs will only make up a small portion of that range compared to unpaired hands.
As we've just covered how few paired starting hands there are, you should be able to guess that unpaired hands will make up the majority of hands in your range. Each unpaired hand has 16 combinations and there are 78 varieties of unpaired hands, meaning there are a total of 1,248 unique unpaired hand combinations.
Unpaired hands are split between four suited and twelve offsuit combinations. There are a total of 312 suited combinations of unpaired hands and 936 offsuit combinations of unpaired hands.
Suited Unpaired hands
There are four suited combinations for each unpaired hand, one for each suit in the deck e.g., , , and . Out of the 1,326 total combinations, suited unpaired hands make up around 23.5% or 312 combinations.
Due to there being only four combos of unpaired hands compared to six combos of pocket pairs and 12 combos of unpaired hands, being dealt AKs is rarer than being dealt AA.
Although suited unpaired hands make up 23.5% of total combinations, they're often played more frequently than offsuit hands so they will make up a higher portion of your opponent's range than 23.5%.
Offsuit Unpaired Hands
Finally, we come to offsuit unpaired hands. These are the most common hand combinations and can be made 12 different ways:, , , , , , , , , , , and .
There are 936 different combinations of offsuit unpaired hands, making up around 70.5% of all possible starting hands.
As offsuit unpaired hands are three times as common as suited hands, these will make up the majority of your opponent's range in the majority of spots.
Card Blockers Computation
If a card is in either your hand or on the board, it's impossible for it to be in your opponent's hand - this is also known as 'card removal'.
As we've covered, there are 6 possible combinations of each pocket pair, with each card appearing in half the combos. This means that if, for example, theis on the flop, the total number of AA combinations in your opponent's range has been halved from 6 to 3 as only the , , and , combinations remain.
If you also hold thein your hand, only one combination of AA is left that your opponent can have - . Then if a third ace comes on the board you know it's impossible for your opponent to have AA.
The first card reduces the total remaining combos by 3, the second card reduces the total remaining combos by 2, and the third card makes a pocket pair impossible.
There are 16 combinations of unpaired hands - 4 suited and 12 offsuit. This time each card appears in a quarter of the total combos (1 suited and 3 offsuit).
However, this time the math isn't as straightforward as the number of combos reduces differently depending on which cards are removed.
For example, if theand are removed then the number of combos is reduced by 8 (4 for each ace) and there are 8 combos ( , , , , , , , ) remaining. However, if the and are removed then the number of combos is reduced by 7 and 9 combos are remaining ( , , , , , , , , )
The easiest way to figure out how many combos are remaining is to treat each card as a side of a multiplication equation.
Without any combos removed there are 4 aces and 4 kings in the deck for a total of 16 combos (4 x 4 = 16). If we remove two of the aces like in the example above, the equation now becomes 2 x 4 which equals 8, and if we remove an ace and a king the equation becomes 3 x 3 which equals 9.
This works for any unpaired hand combination.
Poker Combinatorics and Hand Reading
The main way poker combinatorics is used in poker is to narrow down the possible number of hands your opponent can have in their range. This is done by evaluating what hands your opponent is likely to have in their range, then using information from both your hand and the board to work out the number of combinations that remain.
We've just covered how to work out exactly how many combos of each hand can remain, but how should we be using it in-game?
Wait For Ranges To Narrow
Given how wide preflop ranges are, especially in late position, it's almost impossible to do accurate poker combinatorics when you're early in the hand. There are just too many combos to count and it would take far too long to do at the table.
Instead, you should wait until ranges become narrower and the number of combos there are to count becomes smaller. For example, ranges are often much narrower by the river as there have been three streets of action to gather information on the types of hands your opponent could have. Another example would be a 4bet pot as 4bet ranges are very narrow so it's easy to count how many combos there are.
When trying to count combos it's important to remember that not all combos of hands will be in your opponent's range. For example, if a player is raising from early position, they may have A4s but if they're a good player then they won't have A4o. This means that on a flop such as Q44, there are only two combos of A4 to consider rather than eight.
When trying to count the number of flushes your opponent has in their range, the exact cards that are on the board will have a big impact on that number. The lower the flush cards that are on the board, the more likely your opponent will have a flush.
This is because players will play large suited hands from every position, but will only play lower suited hands from late position. For example, most players will play every suited Ax hand from an early position in a 6-max game. On a three-flush board such as , there are 10 combos of A-high flushes your opponent can have.
However, if the board is instead, your opponent can no longer have Axss in their hand - drastically reducing the number of flushes in their range.
Post Flop Combinations
Let's look at a couple of examples of how you use poker combinatorics in-game:
Example 1 - Preflop All In
In this example, we're 100bb deep in a $1/$2 cash game. We've raised to $5 on the BTN with , our opponent in the SB 3bets to $20, we 4bet to $45 and our opponent goes all-in for $200 total.
We think our opponent is a good, aggressive player and therefore think he will go all-in with TT+ and AK. It's $155 for us to call, what should we do? Let's count the combos.
We have 80% equity against TT and JJ, so that's 12 combos we're beating. It's a tie against the other combo of QQ. We only have 20% against AA and KK, so that's 12 combos we're losing to. Finally, we're a slight favorite against AK which is 16 combos.
Counting all those up there are 12 combos we beat and 12 we lose to so they cancel each other out. As does the 1 remaining combo of QQ. We're around a 55% favorite against the remaining combos of AK. Given the amount of equity we have, it's an easy call.
Example 2 - Facing A River Raise
In this example, we've raised to $5 from the BTN with, 100bb deep in a $1/$2 cash game, and got called by our opponent. The flop is , we bet $7, and our opponent calls. The turn is the , we bet $18, and our opponent calls. The river is the , we bet $35 and our opponent goes all-in for $170 total. What do we do?
We need to think about the range of hands our opponent will raise and the way the action has gone it's reasonable to assume that they're only raising with a flush for value. Let's count the combos of flushes.
We have thein our hand which limits the flush combos our opponent can have. They also didn't 3bet so that will limit their range even more as I expect them to 3bet QJs/JTs almost every time. That leaves , , , , , , , , and - or 9 combos.
Now we need to figure out how often our opponent is bluffing. There are 3 combos of 76s which is the most obvious bluffing hand. It's hard to find other hands that would call on the flop and turn and then bluff the river (aside from turning a pair into a bluff) so let's assume they're only bluffing the 3 combos of 76s.
If we call then we win a quarter of the time. We need to call $135 to win $261, meaning we're getting odds of 1.91:1 and need 34% equity to call. As we only have 25% equity against our opponent's raising range, we should fold.
Using Poker Combinatorics With Player Styles
The type of poker player you're playing against will have an impact on the number of combos you need to consider when thinking about their range. The tighter a player is the fewer combos they're likely to have - and vice versa for loose players.
Tight-aggressive poker players are positionally aware and won't raise garbage hands just for the hell of it. They'll play a tight, solid range from all positions so when you're considering their range on a later street it's safe to assume they won't have any garbage hands in their range.
Expect them to have mostly broadway and Ax hands and not very many combos of lower cards. Be aware that on low boards they're likely to have a lot more no pair hands than they are pairs so it will be easy for them to start over-bluffing without realizing.
Loose-aggressive poker players are harder to range than tight players as they'll play a lot more hands from every position. With these players, you'll need to take in as much information as you can from earlier streets to narrow their range to the point where you can count combos.
They're a lot more likely to play all parts of the deck so they'll be well protected on all board types. However, playing a wide range makes it hard to balance bluffs correctly so try to take advantage of that on disconnected boards.
People tend to say that "I don't like to play against fish as you don't know what they'll show up with". While this is true, they can show up with anything, this is usually said by bad players who don't adapt to their opponent.
To best exploit a fish you need to gauge how aggressive or passive they are. If they're aggressive then they're playing far too many hands to balance bluffs and value bets so calling them down will be profitable. If they're passive then their checking range will contain a lot of 'air' hands so bluffing often when they check will be profitable.
Combinatorics In Poker: FAQ
We've collected and answered the most common questions around poker combinatorics.
What is poker combinatorics?
Poker combinatorics is the use of the mathematical concept of combinatorics to count the number of hand combinations in your opponent's range. It takes into account known cards to narrow down the number of hands, allowing you to make an informed decision on your hand vs your opponent's range.
How many hand combinations are there during pre-flop?
There are 169 individual starting hands in poker, of which there are 1326 unique combinations. There are 78 combinations of pocket pairs, 312 combinations of suited unpaired hands, and 936 combinations of offsuit unpaired hands.
What are blockers?
'Blockers' is a concept which states that if a card is in your hand or on the board then it's impossible for your opponent to have in their hand, therefore 'blocking' them from having certain hand combinations.
How can poker combinatorics give you more information on your opponents?
Poker combinatorics can give you more information on your opponent's range by giving you a mathematically sound way of narrowing it down. Rather than having to guess at how often your opponent is bluffing/value betting, combinatorics allows you to put a number on each part of your opponent's range.
Can poker combinatorics help you win every game?
There is no 'secret formula' that will let you win every time at poker. No matter the amount of skill involved there is still an element of chance so you won't win every time you play. However, using combinatorics and any other poker skills will give you an edge that will help you win more often.
It takes a lot of practice to be able to use poker combinatorics in-game but knowing how to do so will give you a clearer image of your opponent's range.