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Hand CombinatoricsHand Combinations  The BasicsHand 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 stepbystep. 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 2object subsets we can form out of A; that is, the number of combinations of 2object 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 C = n! / x! * (nx)! Plugging in our numbers for set A into this equation, we get: n = 5 C = 5! / 2! * (52)! = 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: 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 52card 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 HandsThere'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:
How we arrive at these numbers is pretty basic:
A Practical ExampleSay 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. One of the top places to play online poker is pokerstars.
