Combinatorics, or counting combos, is a tool that help you dissect your opponent's range. It will give you a better understanding of the nature of other peoples ranges, how often they include draws, made hands or pure bluffs. A combo is simply one specific hand, for instance AcAs (AcAd is a different combo). Before the flop it's 6 possible combos of every pair (3+2+1), four combos of any suited hand (4x1) and 12 combos of all unsuited (and unpaired) hands (4x3).
Post-flop we can find the possible combos of a given unpaired hand by multiplying the reminding cards. If you for instance want to know how many combos there are of AK on a A82 flop, you multiply 3(aces)x4(kings)=12 combos. There are 3 suited (3x1) and 9 (3x3) unsuited combinations left. If we want to compare how often villain will have AK versus AA on this flop, we have to count the combos of AA first, which is 3 (2+1). If villains range were only AK and AA here, we would now know that he is 80% (12/15=0.8) likely to have AK and 20% (3/15=0.2) likely to have AA. By breaking our opponent's range down in this way, it's far easier to figure out the best possible line to take.
To give a few basic numbers of combos:
One pair (AQ on AK2) = 12 combos
Two pair (AK on AK2) = 9 combos
Straight (AK on QJT) = 16 combos
Full house (AK on AAK) = 6 combos
Quads (AK on KKK) = 4 combos
Set (KK on AK2) = 3 combos
Quads (KK on KK2) =1 combo
Remember to not include dead combos if you hold a relevant card in your own hand. If you have AQ on a AQ6 flop for instance, it's just one combo of AA, one combo of QQ and 8 combos of AK left. I think this covers the basics of what combinatorics in poker is about. If we have a good estimate of our opponents range and frequencies, we should be able to relatively fast (after some practice) be able to estimate how big part of it is made up of draws, different made hands and air. For more complex problems, and to get a better feel for different situations, I suggest you download Stox Combo (free). Play a little bit around with it and use it in your session analysis. It works in about the same way as PokerStove, but counts and compare combos rather than calculating equity.
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