Hand Combinatorics

Hand Combinations -- The Basics

Hand combinatorics isn't as scary as it sounds.

It's not some big, complex math technique that only a select few guys with PhDs can understand.

It's an easy, highly useful technique that we poker players can use to help put our opponents on hand ranges.

The technique works kind of like it sounds -- we want to calculate the number of combinations of hands our opponent can be holding in a given scenario.

Let's walk through the process of calculating combinations step-by-step. By the time you're done reading this article, you'll be able to more accurately assess what an opponent's hand range could be.

Combinatorics 101: What's a Combination?

A combination is simply a way to put a set of items together where all the items are drawn from a larger set.

In simple English, a combination is a way to pick some stuff out of a bigger pool of stuff.

Say we have a set of objects A, where A = {1, 2, 3, 4, 5}. We want to figure out the number of 2-object subsets we can form out of A; that is, the number of combinations of 2-object subsets we can pick from the set of objects A.

Calculating this turns out to be pretty simple. All we need to do is follow this simple formula:

n = total # of objects in the set
x = number of items we want to choose
C = combinations of x in set n

C = n! / x! * (n-x)!

Plugging in our numbers for set A into this equation, we get:

n = 5
x = 2

C = 5! / 2! * (5-2)! = 10

So there are 10 unique combinations of 2 items that we might choose from set A.

Now, let's say we have a set of 52 items that we want to draw from. As a basic set of 52 objects, B can be the deck of 52 cards for example.

We just plug the total number of objects in the set (52) into our magic equation, along with the number of items we want to include in each chosen subset (2), and get:
n = 52
x = 2

C = 52! / 2! * (52-2)! = 1326

So there are 1,326 unique combinations of 2 items that we might choose from set B.

You will probably find these numbers to be familiar. If so, good! It's probably because 1,326 is the number of possible starting hands in Holdem poker. What we were acually determining in the above calculation was how many combinations of 2 cards we can pick from a 52-card deck.

Keeping with poker, let's examine a more applicable form of calculating combinations, and some situations in which such a method would be useful.

Combinatorics 101: Combinations of Poker Hands

There's obviously no way you'll be able to whip out a calculator at the poker table and calculate combinations. Luckily, there are some more compact ways of calculating combos that don't require calculators or intense math. We can use these benchmarks to help refine our estimates of opponents' ranges.

There are three basic numbers you need to know:

• There are 12 combinations of any given offsuit unpaired hand in Holdem.
• There are 6 combinations of any given paired hand in Holdem.
• There are 4 combination of any given suited unpaired hand in Holdem.

How we arrive at these numbers is pretty basic:

• There are 4 of any card of a given suit in a deck, and 3 of another particular card of a different suit. So we multiply 4 by 3 to obtain 12, the number of combos for a given offsuit unpaired hand.
• There are 4 of any card of a given suit in a deck, and 3 additional cards left in the deck of that rank. So we multiply 4 by 3 to obtain 12. When counting pairs, there are going to be two ways to make each unique combination of pair; for example, 6c6s and 6s6c. So we must divide 12 by 2 to eliminate double-counts. So 12/2 = 6, the number of combinations of a given pair preflop.
• There are 4 of any card of a given suit in a deck, and only one card after that that can make the suited hand. So (4)(1) = 4, the number of combinations for a given suited hand.

A Practical Example

Say you're playing against an opponent whose range you estimate to be {JJ+, AKo, AQs+} in a given situation. You want to break down the number of hand combinations in his range. You'll calculate the combinations as such:

So there are a total of 44 hand combos in your opponent's range, 24 of which are pocket pairs and 20 of which are unpaired hands.

His range looks a lot scarier when we view it as 4 pocket pair hands and 3 unpaired hands, doesn't it? In reality, it's 50/50 that he's got matching cards in the hole; which can change our equity, and thus our correct action, drastically.

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