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Counting combinations
How many ways each hand can be dealt, how the board and your own cards remove some of them, and why that changes what an opponent holds.
A reference page: an idea the course assumes, without practice or progress. Lessons link here the first time they use the term.
On this page
- Count the combos of a pocket pair, a suited hand and an offsuit hand.
- Remove the combos the board and your own cards make impossible.
- Compare how many value combos and bluff combos a range holds.
- Explain how holding one card changes what the opponent can have.
Not every hand is equally likely
A range is written as hand classes, AA or AK, but the classes are not equally likely. Some can be dealt in more ways than others. Each specific way, such as A♠K♥, is a combo, and counting combos is how you weigh one part of a range against another.
Six, four, twelve
Three counts cover every starting hand:
- A pair: four cards of the rank, and any two of them. 4 × 3 = 12 ordered pairs, and each is counted twice, so 12 / 2 = 6 combos.
- A suited hand: one per suit, so 4 combos.
- An offsuit hand: 4 × 4 = 16 ways to pair the two ranks, minus the 4 suited ones: 16 − 4 = 12 combos.
An unpaired hand like AK therefore has 4 + 12 = 16 combos, against 6 for a pair. In a range of AA, KK and AK, AK is 16 of the 6 + 6 + 16 = 28 combos: more than half of it.
Card removal
A card you can see, on the board or in your hand, cannot be in your opponent's hand. Every card you see removes combos.
For a pair, count the cards of the rank that are left. With one seen, 3 are left: 3 × 2 = 6, halved, is 3 combos. With two seen, 2 are left: 2 × 1 = 2, halved, is 1 combo. For an unpaired hand, multiply the cards left of each rank: with one ace seen, AK has 3 × 4 = 12 combos.
On K♦7♣2♥, each set has 3 combos left, so there are 3 + 3 + 3 = 9 sets in total.
How often is he bluffing? Count two groups
Counting a whole range feels overwhelming, and you rarely need to. When your opponent bets, sort his likely hands into two groups, value and bluffs, and count each roughly.
Say his value hands come to 18 combos and his bluffs to 6. Then 6 of his 18 + 6 = 24 betting combos are bluffs: 6 / 24 = 25%. If he bet half the pot, your call needs 25%, so a hand that beats only his bluffs breaks even.
The doorway to blockers
Your own cards change those counts. Holding a card that his value hands need lowers the value count and raises the share of bluffs; holding a card his bluffs need does the opposite.
Counting is how you measure the effect. On K♦7♣2♥, a king in your hand takes KK from 3 combos to 1, removing 3 − 1 = 2 of them. That is the idea behind blockers, which the ranges module builds on.
Rules of thumb
- A pair has 6 combos, a suited hand 4, an offsuit hand 12, an unpaired hand 16 in all.
- Every card you see removes combos: multiply the cards left of each rank.
- To ask how often he is bluffing, count two groups, value and bluffs, and compare the bluff share with your price.
When they break
- Combos only count what is possible. Real players do not play every possible combo the same way, so a count is only as good as the range behind it.
- Rough counts are fine for decisions, but a single removed card matters most when the groups are small.