7 Combinatorics & Blockers

By the end of this chapter you will be able to count the value and bluff combinations in an opponent’s range and adjust your decision for the cards your own hand removes.

Hand reading rests 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. A player who can count those combinations, and who sees how the cards in their own hand and on the board quietly delete combinations from villain’s range, reads hands far more precisely than one working purely on feel.

This chapter teaches you to count. We start with the raw combinatorics of a 52-card deck, show how the board and your hole cards remove combinations, and build that into one of the most powerful concepts in modern poker: blockers. By the end you should be able to look at a river spot and answer, in seconds, “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 about ranges. Counting combos means counting the ways to pick two specific cards from those still available.

Start with the total. A full 52-card deck contains C(52,2) = 1326 distinct starting-hand combos. Every count in this chapter (the 16s, the 12s, the 4s, the 6s) is a slice of that 1326. As a sanity check, the 13 pocket pairs supply 13 × 6 = 78 combos and the 78 distinct unpaired hands supply 78 × 16 = 1248 combos; 78 + 1248 = 1326. If your range counting ever fails to reconcile against 1326, you have miscounted somewhere.

The numbers you must memorize are two anchors, 16 for any unpaired hand and 6 for any pocket pair, plus the 12 + 4 split of that 16 into offsuit and suited:

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 twelve times more likely to hold an offsuit version than any single specific suited version (12 offsuit combos versus 1 for, say, A♠K♠), and three times more likely to hold offsuit than suited overall (12 versus 4). Suited hands are rarer, which is exactly why flushes are powerful and why a single suit-specific blocker swings so much weight.

Key idea

Two anchors, 16 combos for any unpaired hand and 6 for any pocket pair, plus the 12 + 4 split of that 16 into offsuit and suited. Almost all combinatorics flows from these. If you internalize nothing else, internalize this.

A few more counts worth having ready:

A specific suited hand such as A♥K♥: 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 hold that king, so KK drops from 6 combos to 3.
Hand removal (your blockers) affects only villain’s range, and only from your seat. If you hold the A♠, villain cannot have A♠K♠, A♠Q♠, or the nut-flush version of anything in spades. Your opponent doesn’t know this; 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 (three different suits, so no flush draw is possible yet). You hold QJ and you want to know how much top pair (AK, AQ, AJ, AT, and so on) is genuinely in villain’s calling range.

Take AK specifically. Full deck: 16 combos. One ace is on the board, so only three aces remain available, while all four kings are live. Villain’s AK = 3 × 4 = 12 combos. The ace on the board removed exactly four combos (16 → 12), the four that used the now-dead ace.

Now imagine the same A♠8♥3♦ board, except this time you hold A♣Q♦ and want to count villain’s AK. Two aces are now dead from villain’s perspective, one on the board and one in your hand, while every king is still live because you hold no king. Villain’s AK = 2 aces × 4 kings = 8 combos. Your single ace blocker cut villain’s AK from 12 down to 8, a one-third reduction, purely because you hold one of the cards they need. (Hold an ace and a king and you block both ranks: villain’s AK falls to 2 × 3 = 6 combos, a one-half reduction.)

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 stays 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 as plentiful as possible, which means not holding the very cards they would have chosen to bluff with.

Key idea

Block their value when you bet or raise; unblock their bluffs when you call. Be precise about the asymmetry. A value-blocker such as the nut-flush blocker favors raising over a passive call mainly because it adds fold equity (the value you win when a bet makes a better hand fold) and usually comes with little showdown value. Blocking value is not itself bad for a call; if anything, removing value combos makes a bluff-catch better. What actively ruins a call is a card that blocks their bluffs. The nut-flush blocker, then, is the cleanest example of a premium bluff-or-raise card rather than a card that improves a call.

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 made flushes. If you hold a card of that suit, especially the ace, you remove several of villain’s strongest flush combos and make the nut flush impossible for them. Every flush combo you remove cannot call your shove, so the count of “hands that beat me and call” falls and your bluff is likelier to simply take 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, and you want bluffs to be plentiful. So you would rather 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, which is bad for your call. If your hand has no overlap with their natural bluffing candidates, all those bluffs stay live and your call improves.

Common mistake

“I have the ace of spades, so I have a blocker, so I should call.” Backwards, and exactly why depends on the board texture. (1) On a board where the flush has completed, your A♠ removes villain’s made nut flush, part of their value. That argues for a raise, since you have fold equity and credibly represent the nuts, and your ace-high has almost no showdown value to protect anyway. (2) On a board where the flush has missed, there are no made flushes left to remove; instead the A♠ blocks villain’s busted nut-flush-draw bluffs, which hurts a bluff-catch by deleting some of the air you beat. These are opposite textures: the same card removes value on one board and bluffs on the other, never both at once, yet the naive “I have a blocker, so I call” is wrong on both. Always ask precisely which part of their range, value or bluffs, the card removes.

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 (continuation-bet) 3bb into 5.5bb, BB calls. The turn is the 2♠, putting three spades on board and giving you the nut-flush draw (your A♠ plus the three board spades) alongside ace-high. BB checks and you check back, keeping a draw that still has showdown value. The river is the 7♥, which bricks your flush draw and leaves the A♠ as a pure blocker. Final board: K♠9♠4♥2♠7♥. The pot is roughly 11.5bb. BB leads into you for 9bb, about 80% pot.

You hold A♠Q♦, an ace-high that beats none of villain’s flushes but does beat the busted-draw bluffs, so a flat call is a real bluff-catch. The question is whether raising beats calling, with folding as the floor if neither pays. Calling only realizes showdown value against the busted draws; raising adds fold equity on top. Combinatorics decides which is better.

Step 1: what is BB leading for value? An 80%-pot river lead on a flush-completed board (one where a two-card spade flush is now possible) from the caller is usually polarized, meaning it splits into strong value and bluffs with little in between: flushes for value, and missed draws or weak pairs that give up or turn into a bluff. The slice of value your A♠ actually touches is the made spade flushes, so those are what we count below. Villain’s value also holds some sets and two pair (K9, KK, 99) that beat your ace-high and that your A♠ does not block, so the blocker erases only the nut-flush portion of value. Let’s count the flushes.

BB’s flush combos are two-spade hands. The board shows K♠, 9♠, and 2♠, so three spades are dead. The 4♥ and 7♥ remove nothing in spades, so the 4♠ and 7♠ are both still live. Available spades for villain: A♠, Q♠, J♠, T♠, 8♠, 7♠, 6♠, 5♠, 4♠, 3♠, which is ten live spades, before subtracting any you hold.

Step 2: apply your blocker. You hold the A♠. That single card means villain cannot have the nut flush; there is no A♠x♠ in their range at all. A BB who flats pre-flop and calls a flop c-bet typically 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, and you have deleted every one. The flushes that remain are all second-nut-or-worse.

Step 3: what calls a raise? If you raise, mostly flushes can call, and you have removed the best ones, plus the occasional unblocked set or two pair. A villain holding, say, Q♠J♠ now has to call a big raise fearing exactly the nut flush you represent, and many players fold everything but the very top of their flushes facing a raise. Your A♠ does two jobs at once: it removes nut-flush combos from their value, and it makes your nut-representing raise believable. Against the value you do not block (those sets and two pair), the raise simply leans on fold equity.

Step 4: the decision. A♠Q♦ here is a textbook bluff-raise candidate, precisely because of the A♠: the blocker removes their nut flushes and fuels a credible nut-flush story, so raising out-earns a passive bluff-catch. Contrast a hand like A♥Q♥ on the same river. Same ace-high, the same showdown value (both beat the busted draws, both lose to every flush), but now you hold no spade blocker, remove none of their flushes, and have no credible nut-flush story. Stripped of fold equity, A♥Q♥ has no raise; it is reduced to bluff-catch-or-fold, decided purely by the bluff-to-value ratio, and is the clearly weaker spot. Same rank, same showdown value, opposite decision, and the entire difference is blocker and fold-equity value rather than anything you can see at showdown.

One caveat on the opponent: this bluff-raise only earns when villain will actually fold weaker flushes to your raise, the read that fits a thinking or online player. Against a calling station (an opponent who rarely folds once he has a made hand), the fold equity vanishes, and A♠Q♦ collapses back to a check-and-give-up bluff-catch.

Drill

Take the hand above. (a) Without your A♠, list every nut-flush combo villain could hold given the board (A♠ plus each remaining live spade). Count them. (Answer: the nine other live spades are Q♠, J♠, T♠, 8♠, 7♠, 6♠, 5♠, 4♠, 3♠, so there are 9 nut-flush combos, the A♠ paired with each.) (b) Confirm your A♠ removes all nine. (c) Re-run the spot assuming you instead hold A♣Q♣: how many of villain’s flush combos do you now block? (Answer: zero, because your clubs touch none of their spades, which is why A♣Q♣ has no profitable raise and is reduced to a marginal bluff-catch, called or folded depending on villain’s bluff-to-value ratio.)

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

The board runs out and you decide villain’s value is sets: KK, 99, 44. (We deliberately keep two pair like K9 out of the bucket to keep the count clean. If you did add it, K9 suited is only about 2–3 combos with one king and one nine already on the board, which would nudge value up to ~14–15 and the bluff fraction down a few points; the call below still clears.) Suppose the final board shows one king and one nine.

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, 21 combos in all, your bluff-catcher wins 9/21 ≈ 43% of the time when you call. A pot-sized bet lays you 2:1, so you only need 33% equity to break even. Since 43% > 33%, the call is clearly profitable. Note why: a pot-sized bet should be balanced at exactly 33% bluffs (a 2:1 value-to-bluff ratio), and here villain is bluffing 43%, more than the equilibrium amount. By raw count any polarized betting range holds more value than bluffs, so don’t be fooled by the label “value-heavy”; what decides the call is whether the bluff fraction clears your price, and here it does. Now apply blockers. If your hand removes two of those bluff combos (bad, you’re blocking their bluffs) the call gets worse; if instead you remove 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, since 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 one 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, keep the two removal directions separate. The flopped ace is board removal: it is gone for everyone, cutting villain’s AA from 6 to 3 and their AK from 16 to 12. Your actual blockers are the two kings in your hand. They all but erase villain’s KK (only C(2,2) = 1 combo survives) and trim their AK further, from 12 down to 3 aces × 2 kings = 6 combos. The ace on the board is not your blocker; your kings are.
Barreling rivers. When 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 would 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 for removing 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, or 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.

7.8 Summary

Every unpaired hand is 16 combos and every pocket pair is 6, splitting 12 offsuit and 4 suited; the full deck is 1326 combos.
Board removal subtracts combos for everyone; your own cards (blockers) subtract combos only from villain’s range, and only you know it.
Block their value when you bet or raise; unblock their bluffs when you call.
A card’s showdown value and its blocker value are separate currencies. Weigh both before choosing call, fold, or raise.
Combinatorics only refines a range you estimate from reads, so exact arithmetic on a misjudged range still gives a wrong answer.