Count legal combinations first, then weight them by the line played. Twelve possible bluffs do not mean twelve equally likely bluffs.
Start with the cards available
A pocket pair has six combinations before any relevant card is visible: choose two suits from four. For aces, these are A♠A♥, A♠A♦, A♠A♣, A♥A♦, A♥A♣ and A♦A♣. The order of the two cards does not make a new hand.
Two different ranks have sixteen combinations: four cards of one rank times four of the other. Four are suited and twelve are offsuit. AK therefore has four suited combinations and twelve offsuit combinations before card removal.
Visible cards include your hole cards and the board. Remove them before counting. On an A♥ 9♣ 4♦ flop, with K♠ Q♠ in your hand, an opponent can have three combinations of AA because only three aces remain. AK has three available aces times three available kings, or nine combinations. Of those, A♣K♣ and A♦K♦ are suited; the other seven are offsuit.
| Hand class | Calculation | Combinations |
|---|---|---|
| Pocket pair | 4 × 3 ÷ 2 | 6 |
| Two different ranks | 4 × 4 | 16 |
| Suited non-pair | One per suit | 4 |
| Offsuit non-pair | 16 − 4 | 12 |
Convert a possible range into a weighted range
Imagine a river review where you have identified twelve legal value combinations and twelve legal bluff candidates after card removal. Your bluff-catcher loses to all the value hands and beats all the bluffs. If every candidate bets every time, half the betting range is bluffs.
Now suppose all twelve value combinations bet, but each bluff candidate bets only one-quarter of the time. The effective bluff weight is 12 × 0.25 = 3. The total betting weight is 12 + 3 = 15, making the bluff share 20%, not 50%. You have not removed legal hands; you have modelled their frequency.
Against a half-pot bet, the call threshold is 25%. This weighted example leaves the bluff-catcher short. If the bluff candidates instead bet half the time, their weight becomes six and the bluff share rises to 6 ÷ 18 = 33.33%. That is enough under a no-rake, no-tie, final-call model. The calculation exposes which assumption changes the decision.
Frequency matters as much as possibility
12 value × 100% = 12 weighted combinations
12 bluffs × 25% = 3 weighted combinations
Bluff share = 3 ÷ (12 + 3) = 20%
Check the range for impossible and duplicate hands
Use exact cards when a result is close. “Suited broadways” and “missed flush draws” can include the same hand, so adding both category totals can count it twice. List the combinations once, then label the roles they play.
Check the board as well as your own cards. A hand containing a board card is impossible. On paired boards, a flush may lose to a full house; “made flush” is not automatically the top of a range. A combination’s value or bluff label depends on what it beats and what it expects to be called by.
Do not remove a hand merely because you would have played it differently. Distinguish an impossible hand from an unlikely line. The former has zero weight; the latter may need a small but non-zero weight.
Build a range you can revise
Record the positions, effective stacks, board, sizes and action. Start from a plausible preflop range, then narrow it street by street. Keep the assumptions next to the range so another reader can see why a hand remains.
For uncertain tendencies, compare a conservative, middle and aggressive betting frequency. If the best response changes across those cases, describe the decision as sensitive to the read. Do not disguise that uncertainty with an exact-looking percentage.
Combination counting improves the structure of your thinking. It does not reveal the opponent’s cards, establish a solver strategy or prove that a small sample of observed bluffs will continue.
Pause and make your decision.
One queen is on the board and you hold another queen. How many combinations of QQ can an opponent have? If eight bluff candidates bet 50% of the time against twelve value combinations that always bet, what share of the betting range is bluffs?
For your next study session: Write out one river range using exact suits. Remove unavailable cards, eliminate duplicates, and attach a betting frequency to each category.
Put the idea to work.
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