A Beginner’s Guide to Poker Combinatorics

Combinatorics is a fancy term for evaluating the number of possible “combinations” (combos) of any given hand: the combination of 2 cards of certain ranks and suits. It allows us to answer questions like how many different versions of AK you can hold in a specific spot, what hands make for better bluff-catchers and many more.

Combo Counting Basics

The Hand Matrix

There are 1326 possible combinations of starting hands in Texas Hold’Em. They are frequently represented by a grid that is commonly referred to as a hand matrix.

This leads us to the conclusion that there are 78 pocket pairs (6 combos times 13 squares), 312 suited hands (4 combos times 78 squares) and 936 unpaired, offsuit hands (12 combos times 78 squares).

The Card Matrix

Dead Cards

The above ratios are applicable preflop but falter when in the presence of “dead cards”. Dead cards refer to cards that are known to not be in a player’s hand. The most common example occurs when a flop is dealt. The flop cards are dead as it is impossible for any player to hold them.

The simplest way to count combinations is to multiply the number of unseen cards. Let’s go over some examples to familiarize you:

BTN opens, BB calls, flop is AK2:

Other Card Removal Examples

Dead cards exist when public information makes certain hands impossible to hold. There exist other, more subtle versions of card removal, however. One notable version is blockers. This is when one player uses the private information provided by their hand, or private dead cards if you will, to combinatorically narrow their opponent’s range.

Simple Blocker Example

Blockers are a vital part of poker strategy that can make or break close decisions. By eliminating specific combos from players’ ranges, you can shift their action frequencies and value to bluff ratios.

Let’s examine a simple example where we can make use of this effect:

🎲 [6Max Cash, 500NL, 100BB]

BTN opens to 2.5BB and BB defends. BB check-calls a 33% flop c-bet, a 125% turn c-bet and faces an 85% river c-bet on J♠5♥2♠K♦8♠. Which of the following hands can profitably call?

Strategic vs Actual Frequencies

Despite blockers changing range action frequencies, solvers (including GTO Wizard) do not display the impact of this effect in their strategic frequencies. This leaves a discrepancy between strategic and actual frequencies. This is clearly demonstrated on flush boards:

[6Max Cash, 500NL, 200BB]
HJ opens to 2.5BB and BB defends. BB check-calls a 130% c-bet, the turn checks through and BB check-shoves (for 407%) vs a 61% river bet on a board of Q♠J♠4♥7♦2♠.

Despite the strategic frequency window declaring that HJ calls 19.8% of the time (1.07 combos out of a total 5.51 combos), they only call 13.8% in reality (0.52 combos out of a total of 3.78 combos).

The 19.8% frequency includes “ghost” combos: ones that cannot exist due to our opponent’s blockers.

BB only shoves hands that contain the A♠, however, HJ’s 19.8% calling frequency includes many A♠ hands. Once removed, the calling frequency becomes 13.8%.

This doesn’t mean the strategy is incorrect. The expected value calculations account for this effect. It’s just a matter of interpreting and displaying data. Do we want to examine the strategic frequency or the actual frequency? Some solvers have a switch that allows you to alternate between both views.

Check out this spot in GTO Wizard.

Card Bunching

The final card removal effect we will examine is card bunching. Players tend to fold low cards more often than high cards preflop. So when many players have folded, the remaining players and deck are skewed towards high cards.

At a 3-handed table, BTN acts first, and every combo in their range is equally likely. This is not the case at a 6-handed table when several players have already folded. This card removal affects the ranges and strategy of all players to act, frequencies, and runout probabilities.

This means that BTN is more likely to hold higher cards when it has folded to them. This can change the BTN’s actual opening frequency even if they don’t change their strategic opening frequency.

Let’s visualize this effect. The following animation displays the density of cards from 2 through Ace in the deck after several players fold. Remember that they’re more likely to fold low cards, so the remaining cards in the deck (and player’s ranges who are yet to act) are more top-heavy.

Conclusion

There are 1326 possible combinations of starting hands in Texas Hold’Em made up of 78 pocket pairs, 312 suited hands and 936 unpaired, offsuit hands. Some of these combos can be blocked by the board or players’ hands and ranges, further decreasing the number of available combos. This should be accounted for and optimized for when forming a strategy.

There are 3 key card removal effects: dead cards, blockers and card bunching; each with their own unique properties.

Try this combinatorics poker puzzle if you feel like challenging your skills!

If there’s one thing you take away from this article, it’s to think about your hands a little less linearly and more in relation to the range you are facing and how this dictates your use of blockers. Good luck at the tables!