Combinatorics (Combos)

Also known as: combinatorics, combos, hand combinations, combo counting

Counting how many specific card combinations make up a hand class, the basis of range construction and bluff/value balance.

Combinatorics is the bookkeeping of ranges: how many distinct two-card combos form each hand. The base counts before any card removal:

These numbers drive range vs range equity and bluff-to-value-ratio: to size value and bluffs correctly you count combos, not hand "types." A board that pairs or shows known cards changes the counts — three of a kind on the board leaves only \(\binom{2}{2}=1\) combo of the top set, and so on.

Combinatorics also exposes blocker effects precisely: holding one ace drops villain's AA combos from 6 to 3 and AK from 16 to 12. When you balance a river value-bet range against bluffs, you're literally counting value combos and bluff combos so the ratio matches your sizing. It's the math under every solver output, done by hand.

Example

Villain's value range on a K♠ 7♥ 2♦ river is {KK, 77, 22, AK}. Sets: KK has \(\binom{2}{2}=1\) (two kings on... none on board, so 6 minus removal — here board has one K, so \(\binom{3}{2}=3\)). With the board K, KK = 3 combos, 77 = 3, 22 = 3, AK = 16. Counting combos like this — not just "four hands" — tells you exactly how many bluffs to add for balance.