7 Combinatorics & Blockers

Hand reading is the soul of poker, but it is built on a foundation of arithmetic. When you say “I think villain has a flush draw or a set here,” you are making a claim about how many specific card combinations support each story. The player who can count those combinations — and who understands how the cards in their own hand and on the board quietly delete combinations from villain’s range — is reading hands in a fundamentally more precise way than the player working purely on feel.

This chapter teaches you to count. We will start with the raw combinatorics of a 52-card deck, show how the board and your hole cards remove combinations, and then build that into the single most practically powerful concept in modern poker: blockers. By the end you should be able to look at a river spot and answer, in seconds and out loud, “there are X combos of value and Y combos of bluffs, and the card in my hand kills Z of them.”

7.1 The raw numbers: how many combos is a hand?

A combo (combination) is one specific two-card holding, identified by suit. A K and a K are different combos even though we lump them together as “AK” when we talk ranges. Counting combos is just counting the ways to pick two specific cards from those that are still available.

The three numbers you must memorize:

Holding type	Combos	Why
Unpaired hand (e.g., AK, QJ)	16	4 aces × 4 kings = 16
— offsuit portion	12	the suited ones removed
— suited portion	4	one per suit (A♠K♠, A♥K♥, A♦K♦, A♣K♣)
Pocket pair (e.g., 99)	6	choosing 2 of the 4 nines: C(4,2) = 6

That 16 = 12 + 4 split matters constantly. When someone “has AK,” they are four times more likely to hold an offsuit version than any single suited version, and three times more likely to hold offsuit than suited overall. Suited hands are rarer, which is exactly why flushes are powerful and why a single suit-specific blocker swings so much weight.

Key idea

Three sacred numbers: 16 combos for any unpaired hand (12 offsuit + 4 suited), and 6 combos for any pocket pair. Almost all combinatorics flows from these. If you internalize nothing else, internalize this.

A few more counts worth having ready:

A specific suited hand, like A♥K♥ specifically: 1 combo.
A pair after one of its cards is dead (e.g., 99 when a 9 is on the board): C(3,2) = 3 combos.
A pair after two are dead: C(2,2) = 1 combo.
An unpaired hand after one of its ranks loses a card (e.g., AK when one ace is gone): 3 × 4 = 12 combos.

7.2 Card removal: the board and your hand delete combos

Here is the pivot from textbook counting to live hand reading. The 16-and-6 numbers describe a full deck. The moment cards appear — on the board, or in your own hand — they are no longer available to villain, and every combo that needed one of those cards vanishes.

This is card removal, and it works in two directions you must keep separate:

Board removal affects everybody equally. If the flop is K-7-2, neither you nor villain can be holding the K♣ if it is sitting on the felt. KK drops from 6 combos to 3.
Hand removal (your blockers) affects only villain’s range from your seat. If you hold the A♠, villain cannot have A♠K♠, A♠Q♠, the nut-flush version of anything in spades, and so on. Your opponent doesn’t know this, but you do, and that asymmetric information is an edge.

Worked count: how many AK combos remain after an ace flops?

You raise pre-flop, the big blind calls, and the flop comes A-8-3 rainbow. You hold Q J and you are wondering how much top pair (AK, AQ, AJ, AT, etc.) is genuinely in villain’s calling range.

Take AK specifically. Full deck: 16 combos. But one ace is on the board, so only three aces remain available. AK is now 3 kings… wait — count carefully: the aces available to villain are 3 (one is on the board), the kings available are 4. So AK = 3 × 4 = 12 combos remain. The ace on the board removed exactly four combos (16 → 12), the four that used the now-dead ace.

Now suppose the turn pairs you up irrelevantly but a second ace is irrelevant — let’s instead say you actually hold A K yourself in a different line and want to count villain’s AK. Now two aces are dead from villain’s perspective (one on board, one in your hand). Villain’s AK = 2 aces × 4 kings = 8 combos. Your single ace blocker chopped villain’s AK from 12 down to 8 — a one-third reduction — purely because you are holding one of the cards they need.

That is the entire mechanism of blockers in one example: a card you hold is a card they cannot have.

7.3 Blockers and unblockers

A blocker is a card in your hand (or on the board) that removes combos from a part of villain’s range you care about. An unblocker is the deliberate absence of such a card — you do NOT hold a card villain needs, so all of that part of their range remains live.

The reason blockers matter is that poker decisions are won and lost at the margins, and combinatorics is how you measure the margin. Two ideas drive everything:

When you bluff, you want to block their value and their calls. Holding a card that removes the hands that would call you means there are simply fewer combos left that can punish your bluff.
When you call (as a bluff-catcher), you want to UNblock their bluffs. You want their bluffing combinations to be as plentiful as possible, and you want to not be holding the very cards they would have chosen to bluff with.

Key idea

Bluffing math wants blockers to value; calling math wants you to unblock bluffs. Said another way: a great card to bluff with is a terrible card to call with, and vice versa. The nut-flush blocker is the cleanest example — it is a premium bluff card and a poor call card.

Why blocking value helps a bluff

Suppose the river completes a flush and you are deciding whether to bluff-shove. The hands that beat you and will call are, say, made flushes. If you hold a card of that suit — especially the ace of that suit — you have personally removed several of villain’s strongest flush combos and you hold the card that makes the nut flush impossible for them. Every flush combo you remove is a combo that cannot call your shove. You have not changed the size of the pot, but you have tilted the count of “hands that beat me and call” downward, which raises the chance your bluff simply takes it down.

Why unblocking bluffs helps a call

Now flip seats: villain shoves the river and you hold a bluff-catcher — a hand that beats a bluff but loses to any value. Whether you call profitably depends on the ratio of bluff combos to value combos in villain’s range. You want bluffs to be plentiful. So you want to not hold the cards villain would have bluffed. If a busted straight draw is the likely bluff, and you hold cards that block that draw, you have removed villain’s bluffs — bad for your call. If instead your hand has no overlap with their natural bluffing candidates, all those bluffs remain live, and your call improves.

Common mistake

“I have the ace of spades, so I have a blocker, so I should call.” Backwards. The nut-suit blocker removes villain’s value (their flushes), which is an argument to bluff, not to call. When you are facing a bet and considering a call, the blocker you want is one that removes villain’s value — and the nut-flush blocker often does the opposite of what loose-thinking players assume, because it also blocks the very flush draws they might be bluffing with. Always ask precisely: which part of their range does this card remove?

7.4 A fully worked river example

Let’s put the whole machine together on a single hand.

Setup. 100bb effective, online 6-max cash. You open the CO to 2.5bb with A♠Q♦, the BTN folds, the BB calls. Heads-up to a flop of K♠ 9♠ 4♥. BB checks, you c-bet 3bb into 5.5bb, BB calls. Turn is the 2♠, completing the front-door spade flush. BB checks, you check back (you have ace-high with the nut-flush blocker but no made hand). River is the 7♥. Final board: K♠ 9♠ 4♥ 2♠ 7♥. Pot is roughly 11.5bb. BB leads into you for 9bb — close to a pot-sized bet.

You hold A♠Q♦. You have ace-high. You cannot win at showdown against anything that bets for value. The question is whether to fold or raise as a bluff (calling is pointless — ace-high beats nothing that bets here). Combinatorics decides it.

Step 1 — what is BB leading for value? A blocking-sized-then-jammed line aside, a pot-sized river lead on a four-flush board from the caller is usually polarized: flushes for value, and missed draws / weak pairs giving up or turning into a bluff. The value class is made spade flushes. Let’s count them.

BB’s flush combos are two-spade hands. The board shows K♠, 9♠, 2♠ — three spades are dead. The remaining spades are: A♠, Q♠, J♠, T♠, 8♠, 7♠… but 7 is now a heart on the board, so 7♠ is still live; the 7♥ removes nothing in spades. Available spades for villain: A♠, Q♠, J♠, T♠, 8♠, 7♠, 6♠, 5♠, 3♠ — and we must subtract any you hold.

Step 2 — apply your blocker. You hold A♠. That single card means villain cannot have the nut flush — no A♠-x♠ in their range at all. Realistically a BB flatting pre-flop and calling a flop c-bet holds suited spade combos like Q♠J♠, J♠T♠, T♠8♠, 8♠7♠, suited connectors and a few suited broadways. Without your A♠, the nut-flush combos (A♠ plus another live spade) would be a meaningful chunk of their value. You have personally deleted every one of them. The flushes that remain are all second-nut-or-worse — and crucially, you hold the card that lets you credibly represent the nut flush yourself.

Step 3 — count the bluff-catcher math from their side, then yours. If you raise, what can call you? Only flushes — and you have removed the best ones. A villain holding, say, Q♠J♠ now has to call a big raise on a board where the A♠ is conspicuously absent from the action, fearing exactly the nut flush you are representing. Many players fold everything but the very top of their flushes facing a raise. Your A♠ blocker does two jobs at once: it removes nut-flush combos from their value, and it makes your bluff-raise representing the nuts believable.

Step 4 — the decision. A♠Q♦ here is a textbook bluff-raise candidate, not a call and not a pure fold, precisely because of the A♠. Contrast it with a hand like A♥Q♥ on the same river: same ace-high, but now you hold zero spade blockers, you remove none of their flushes, and you have no credible nut-flush story. A♥Q♥ should simply fold. Same rank, same showdown value, opposite decision — and combinatorics is the only thing that distinguishes them.

Drill

Take the hand above. (a) Without your A♠, list every nut-flush combo villain could hold given the board (A♠ + each remaining live spade). Count them. (b) Now confirm your A♠ removes all of them. (c) Re-run the spot assuming you instead hold A♣Q♣: how many of villain’s flush combos do you now block? (Answer: zero — your clubs touch none of their spades — which is why A♣Q♣ has no business raising and should fold.)

7.5 Counting a full range: a practical method

You won’t always have a solver. Here is a hand-readable procedure for any river bet-or-call decision:

Define villain’s two buckets: which combos are value (beat you, will get the money in) and which are bluffs (you beat them, they’re betting anyway).
List the holdings in each bucket as hand classes (e.g., “sets: KK, 99, 44”).
Convert each class to a raw combo count using 16 / 12 / 4 / 6.
Subtract board removal — kill any combo using a card already on the board.
Subtract your blockers — kill any combo using a card in your hand.
Compare the surviving totals against the pot odds you’re being laid.

Example: counting value vs. bluffs

Board runs out and you decide villain’s value is sets and two-pair: KK, 99, 44, plus K9s-type two pairs. Suppose on the final board one king and one nine are present.

KK: a king is on the board, so 3 kings remain → C(3,2) = 3 combos.
99: a nine is on the board → C(3,2) = 3 combos.
44: no four on the board (say) → full 6 combos.
Total sets ≈ 12 combos of value.

Now the bluffs. Say the missed draws are J♠T♠, Q♠J♠, T♠8♠ style busted spade draws plus a couple of busted straight draws. Each specific suited combo is just 1 combo, so if you can name eight or nine plausible busted-draw combos, that’s roughly 8–9 bluff combos.

With ~12 value and ~9 bluffs, villain is value-heavy: bluffs are about 43% of the betting range, so a bluff-catcher needs better than roughly that to call a pot-sized bet (which lays you 2:1 and requires you to be good ~33% of the time — here you clear it). Now apply blockers: if your hand holds a card that removes two of those bluff combos (bad — you’re blocking their bluffs) the call gets worse; if instead you hold a card removing a set combo (good — you’re blocking value) the call gets better. This is the whole game: you are constantly nudging a ratio by one or two combos, and one or two combos is frequently the entire margin between a profitable call and a losing one.

Common mistake

Counting a specific suited combo as if it were 4 combos. J♠T♠ is one combo, not four. Suited reads are powerful precisely because they are rare — when you put villain on “a spade draw,” you are putting them on a thin slice of combos, and a single blocker can erase a large fraction of it. Conversely, do not under-count offsuit value: AK offsuit is 12 combos and quietly dominates ranges.

7.6 Blockers in other spots

The river bluff-catch is the marquee application, but combinatorial thinking runs through the whole game tree:

Pre-flop 3-bet bluffs. Hands like A5s and A4s make popular light 3-bets partly because the ace blocks villain’s AA and AK combos (removing 1 ace cuts AA from 6 to 3 and AK from 16 to 12), reducing the chance you run into a 4-bet or a flat that crushes you — while the suited wheel card gives you playable equity when called.
Set-mining and “is my big pair good?” When you hold KK and an ace flops, you have removed exactly one ace from villain’s AA and AK, a small but real reduction in the combos that have you beat.
Barreling rivers. Choosing which missing-draw hands to triple-barrel as bluffs: prefer the ones that block villain’s calling range (e.g., a busted draw that also holds a card to the nut straight or flush they’d call with) over the ones that block nothing.
Calling down vs. a maniac. Against someone who over-bluffs, you want maximum unblocking — call with the bluff-catchers that leave all of their air in the deck.

Key idea

A holding’s showdown value and its blocker value are two different currencies. Ace-high with the nut-flush blocker has near-zero showdown value but high blocker value (great bluff). A small set has huge showdown value but may block none of villain’s bluffs (great value-bet, mediocre bluff-catcher to remove air). Always evaluate both before you act, and let the decision — call, fold, or raise — follow the currency that actually matters in that spot.

7.7 Honest caveats: combos are a model, not a certainty

Everything above assumes you have correctly defined villain’s range — and that is the soft, human part of the equation. Combinatorics gives you exact arithmetic on top of an estimate. If your read on their value bucket is wrong, perfectly counted combos will lead you confidently to the wrong answer. Population tendencies (most low-to-mid-stakes players under-bluff rivers; many over-fold to raises representing the nuts) are typical ranges, not laws, and any individual villain can deviate wildly.

So treat combo counting as a discipline that sharpens reads you build elsewhere — from bet-sizing tells, timing, player type, and history — rather than a replacement for them. The strongest players do both at once: they form a range from behavioral reads, then count combos to translate that range into a precise fold/call/raise. Get fluent enough that the counting is automatic, and your attention stays free for the parts of poker that can never be reduced to arithmetic.

Drill

For one full session, before every river decision say two numbers out loud (in your head if live): “value combos / bluff combos.” Don’t even act on them differently at first — just build the habit of generating the count. Within a few sessions the relevant blockers in your own hand will start jumping out at you automatically, and you’ll notice the spots where one card flips the decision.