Combinatorics: An Introduction to the Study of Card Combinations

Dave Roemer | 3 weeks ago in Poker Theory and Concepts

The purpose of this article is to introduce you to the concept of combinatorics… that is, card combinations and how they apply to poker.

Combinatorics literally refers to the number of card combinations a player can have. For example, how many ways can a player hold pocket aces? That is, how many combinations of aces are there? When we group the pocket aces by suits we find there are 6 possible ways to hold that hand (6 combinations):

A♥A♦, A♥A♣, A♥A♠, A♦A♣, A♦A♠, A♣A♠

Let’s take a couple basic examples of how one might use this to idea in a poker situation to illustrate.

Example 1:

You open for a raise with TT. A very tight player reraises you all in. You believe their range of hands to do this is specifically JJ+ and AK. What is the chance they have AK?

As we showed above, there are 6 ways to hold a pocket pair. What about AK?

There are 4 combinations of AK suited: A♣K♣, A♦K♦, A♠K♠, A♥K♥
There are 12 combinations of AK off suit: A♣K♠, A♣K♥, A♣K♦, A♠K♣, A♠K♥, A♠K♦, A♥K♣, A♥K♠, A♥K♦, A♦K♣, A♦K♠, A♦K♥

So there are 40 total card combinations in a range of JJ+/AK:

6 combos each of JJ/QQ/KK/AA = 24 combos of pairs
AK (both suited and off suit) = 16 combos

You will be up against AK specifically 40% of the time then (16/40), and up against an overpair 60% of the time, making you more likely to be a 4-1 underdog in the hand than a small coin flip favorite.

Example 2:

An opponent open raises approximately 8% of starting hands. This is roughly 77+, ATs+, AJo+, and KQs. If we 3-bet shove over their open raise, we believe they will only call our shove with TT+, AK, and AQ, and fold everything else. How often is our shove getting called?

The opening range above is actually 7.8% of starting hands. Their shove calling range is 4.7% of all starting hands. So we can expect our shove to be called 4.7/7.8 = .6 or approximately 60% of the time.

This can be expressed combinatorically as well:

Opening Range:

77-AA = 48 combos (6 for each pair)
AK = 16 combos
AQ = 16 combos
AJ = 16 combos
ATs = 4 combos
KQs = 4 combos
Total hand combinations in opening range = 104

Shove calling Range:

TT-AA = 30 combos
AK = 16 combos
AQ = 16 combos
Total hand combos in shove calling range = 62
So he is calling our shove with 62 combos/104 opening combos = .596 or approximately 60% of the time.

Blockers

You will often here the term “blockers” used in relation to shoving ranges, calling ranges, and the like. A blocker is a known card that the opponent can’t hold, thus reducing their combos of particular holdings that would use that card.

Going back to the 6 combinations of AA we listed earlier, you may notice that each suit is represented 3 times (naturally, as it pairs once with each other suit). So what happens if we have an “ace blocker” in our hand? For example, if we hold the ace of diamonds, then we know our opponent can’t have it, so now there are only 3 possible ways they can hold AA instead of 6.

The practical application of blockers lies in the removal of certain hand possibilities or combinations from an opponent’s range.

Going back to Example 2 above, we expected to get called 60% of the time on our all in. Notice we didn’t address what hand we were re-raising all in with? How do the numbers change if our holding is AcKs?

Opening Range:

77-QQ = 36 combos (6 for each pair)
KK-AA = 6 combos (3 for each, as we know 1 ace and 1 king can’t be in their hand)
AK = 9 combos
AQ = 12 combos (Removing the Ac leaves 3 AQs combos and 9 AQo combos)
AJ = 12 combos
ATs = 3 combos
KQs = 3 combos
Total hand combinations in opening range = 81

Shove calling Range:

TT-QQ = 18 combos
KK-AA = 6 combos
AK = 9 combos
AQ = 12 combos
Total hand combos in shove calling range = 45

So now, due to the removal of an ace and king from the strong hand possibilities they can held, they will call our shove 45/81 times or approximately 55.5% of the time, not an insignificant difference from the 60% of the time we’d be called if we didn’t hold any blockers to AA, KK, and AK.

Combinatorics provides us a basic way to quantify how ranges are constructed in terms of the possible number of ways to have certain holdings. By using this tool away from the tables, along with the concept of blockers, you can break down common situations and gain a stronger familiarity with how ranges come together in a more realistic and practical way that mere feelings or hunches.

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