Poker Combinatorics: Counting Combos in Your Opponent's Range

Poker combos start with three counts: a pocket pair has 6 combos, a suited hand has 4, and an offsuit hand has 12, before card removal. Remove every combo containing a board card or one of your own cards. In our teaching river spot, you hold Q♣J♠ and face a $45 bet into $45: the required bluff share is 33.3%. Forgetting your cards gives 32.8% and a fold; removing Q♣ and J♠ gives 37.5%, clearing the threshold for the stated range. But removing 87s from the bluff set takes away 4 combos and drops the share to 31.8%; a $90 bet also makes the call fail. The margin is only 3 bluff combos.

A “combination” in the sense of a hand rank, such as a pair, straight or flush, is a different question. Here, a combo is one way to deal two specific cards, such as Q♣Q♥; QQ is a hand class containing several combos.

How many combos does each hand have?

Without known cards, each pocket pair has 6 combos, each suited hand class has 4, and each offsuit hand class has 12. In range notation, s means suited and o means offsuit.

Across the deck: 13 × 6 + 78 × 4 + 78 × 12 = 78 + 312 + 936 = 1,326 combos, or C(52, 2).

These occupy 169 hand classes: 13 pairs, 78 suited classes and 78 offsuit classes. See preflop range charts for how to read the 13×13 grid and a range percentage.

AK without a suit qualifier includes both AKs and AKo: 4 + 12 = 16 combos. If every combo of QQ and AK has equal weight, their shares are:

• QQ: 6 of 22 combos, or 27.3%.
• AK: 16 of 22 combos, or 72.7%.

Counting them as “one hand each” misses the difference between hand classes and physical combos.

How do you remove known cards?

Remove every combo containing a board card or one of your hole cards. These are dead cards for counting your opponent’s possible hands.

For a pocket pair, with n live cards of that rank:

Combos = n × (n − 1) / 2.

For two different ranks:

• All combos = live cards of the first rank × live cards of the second.
• Suited combos = the number of suits in which both cards remain live.
• Offsuit combos = all combos minus suited combos.

For example, Q♦ on the board leaves three queens: QQ = 3 × 2 / 2 = 3 combos. If you also hold Q♣, only two queens remain: QQ = 2 × 1 / 2 = 1 combo.

The last six rows isolate Q♣, without board removal. It removes A♣Q♣ and offsuit combos containing Q♣, but leaves hand classes without a queen unchanged.

Unlike combos, which count your opponent’s possible hands, outs and the rule of 2 and 4 count cards in the deck that can improve your hand.

How do blockers cut both ways?

In our river spot, Q♣ removes 11 value combos, reducing the board-only count from 41 to 30. J♠ also removes bluffs: JTs and J8s each fall from 4 to 3, reducing the total from 20 to 18. The balance improves for calling this range because your cards remove more value than bluffs, not because a blocker is inherently helpful.

How do combos translate into a river decision?

For the stated range and $45 bet, removing Q♣J♠ changes the result from below the 33.3% threshold to above it: 32.8% becomes 37.5%. That result is sensitive to the bluff set: no 87s means 4 fewer bluff combos and a 31.8% share; against a $90 bet, even the original 37.5% is insufficient.

The spot is 6-max cash with $1/$2 blinds, $200 effective stacks and no rake. You are BB with Q♣J♠.

You have top pair with a jack kicker and $178 behind, so the river bet fits the stack.

Assign BTN this betting range:

• Value that beats Q♣J♠: QQ, 99, 66 and 44 are sets; AQs, AQo, KQs and KQo are top pair with a better kicker; Q9s and 96s are two pair.
• Bluffs that lose to Q♣J♠: JTs, T8s, J8s, A5s and 87s are missed draws.
• All other hands are outside the calculation.

Give every listed combo equal weight and assume BTN bets each one. Every value combo beats your hand, every bluff loses, and there are no ties.

This range is a teaching model, not field data or solver output.

Apply the same removal process to the bluffs:

Teaching diagram: value combos 64, 41 and 30, bluff combos 20, 20 and 18, bluff share 32.8% and 37.5% against a 33.3% call thresholdTeaching diagram for the river Q♦9♦4♣2♠6♥: hero holds Q♣J♠ and faces a $45 bet into a $45 pot. Grid first, then board, then hero's cards. It is a conclusion from the stated teaching range, not an observation of play.

Required equity is bet / (pot before the bet + 2 × bet) = 45 / 135 = 33.3%; this matches C/(S+C) in pot odds and implied odds, where C is the call and S includes the opponent’s bet. To check it in the pot odds calculator, enter 45 for the pot before the bet and 45 for the bet.

Here, bluff share equals your equity because you beat every bluff, lose to every value hand and cannot tie. More generally, bluff share describes the composition of a betting range; it is neither an observed statistic nor a synonym for equity.

Calling $45 can win $90: the original $45 pot plus BTN’s $45 bet.

EV of calling = bluff share × (pot + bet) − (1 − bluff share) × bet.

EV is zero at the break-even threshold, displayed as 33.3%. Percentages below are rounded to one decimal place and money to cents; the calculations use unrounded shares.

• Counting error, removing only the board: 41 value combos and 20 bluffs give 32.79%, displayed as 32.8%, and calling EV of −$0.74. The call fails the threshold.
• Remove the board and Q♣J♠: 30 value combos and 18 bluffs give 37.50%, displayed as 37.5%, and calling EV of +$5.63. The call clears the threshold for this range and bet.

The margin is only 3 bluff combos: 18 compared with a break-even 15 against 30 value combos. This example teaches the counting method, not a call to copy on a similar board.

What changes the conclusion?

Bet size, the bluff set, value composition and bluff frequency can each change the decision independently. Your cards are removed throughout this table; each row changes the base case separately.

Effective combos in the last two rows are physical combos weighted by their inclusion frequency, not fractional remaining hands.

Against a pot-size bet with 30 value combos, BTN must bet at least 83.3% of the original bluff combos to reach break-even. The call clears the threshold only with enough bluff weight: a $90 bet, no 87s, adding 22, Q9o and 96o together, or betting only 80% of the bluffs each independently changes the result to a fold.

How can you repeat the count in Range Builder?

Range Builder verifies 64 → 41, but you must calculate the final 41 → 30 manually: the tool removes board cards, not your hole cards.

1. Open Range Builder.
2. Enter QQ,99,66,44,AQs,AQo,KQs,KQo,Q9s,96s in "Range notation" and click "Apply to grid". Keep the default action "Raise". "Your range" shows 64 combos and 10 hands (hand classes), 4.8% of 1,326.
3. Set Q♦9♦4♣2♠6♥ in "Board (optional)".
4. In "How your range hits the board", read "Across 41 combos in range (cards on the board are removed)."
5. Subtract Q♣ and J♠ by hand using the formulas: the value range falls from 41 to 30.

After you click "Apply to grid", the field rewrites the list in compact form: QQ, 99, 66, 44, AQs, KQs+, Q9s, 96s, AQo, KQo+. Here KQs+ means only KQs and KQo+ only KQo, because Q is already the highest kicker under K; the set is the same 10 hand classes and 64 combos.

Range Builder on a phone: after Apply to grid the field shows QQ, 99, 66, 44, AQs, KQs+, Q9s, 96s, AQo, KQo+ with 64 combos, and the board Q♦9♦4♣2♠6♥ leaves 41 combosReal Range Builder interface on the live ToolsGambling site (2026-10-17, 390 px wide), values typed by hand. The tool removes only the board, so subtract hero's Q♣ and J♠ yourself.

The "combos" count in "Your range" is before board removal. Read it separately from the board-adjusted count.

For the bluffs, JTs,T8s,J8s,A5s,87s gives 20 combos in the grid and 20 after board removal, because the board contains no J, T, 8, 7, A or 5. Manually removing J♠ leaves 18.

Counting combos does not replace checking equity. Once you have defined the opponent’s range, check your hand’s equity against it separately in the Equity Calculator.

What counting routine should you follow?

Define the betting range, remove dead cards, apply betting frequencies and then compare the result with the price of calling.

1. Write down the opponent’s range. Separate hands that beat yours from hands you beat, and check for ties.
2. Convert hand classes into combos. Start with 6, 4 and 12 rather than counting grid cells.
3. Remove the board and your cards. Exclude every physically impossible combo.
4. Apply betting frequencies. Weight combos according to how often they enter the betting range.
5. Calculate bluff share and check equity. Establish which hands win, lose or tie before relating the two.
6. Compare with the threshold and test sensitivity. Change the bet size, bluff set and value set to see how much room the result has.

Avoid these counting mistakes:

• Counting classes instead of combos. QQ and AK carry different weights: 6 versus 16 before removal.
• Forgetting your own cards. In this example, that error changes which side of the calling threshold the count reaches.
• Giving equal weight to combos that act differently. A possible holding need not enter the betting range at full weight.
• Using a share of 1,326 instead of live betting-range combos. The river decision depends on the range remaining after removal and weighting.
• Treating a blocker as a guarantee of a good bluff. It can remove strong hands and hands that might fold; consider both effects.