Poker Combinatorics: Counting Combos at the Table

A deck deals 1,326 distinct starting hands: six combinations of every pocket pair, four of every suited hand, twelve of every offsuit hand. Combinatorics is the skill of counting how many of those combinations an opponent can still hold once you subtract the board and your own two cards. It converts a vague read like "he could have a set" into a countable one — nine combos of sets against thirty-two combos of missed draws — and only the second version is a decision you can act on.

Where 1,326 comes from

There are 52 ways to deal the first card and 51 ways to deal the second, which gives 2,652 ordered pairs. Order does not matter in hold'em, so halve it. You get 1,326 hands.

Split that total by hand type and the three numbers you need for everything else fall out.

• Pocket pairs take two cards from the four of a rank, which is six combinations each. Thirteen ranks gives 78 pair combos.
• Unpaired hands match any of four cards with any of four others: sixteen combinations. Four of those are suited and twelve are offsuit.
• The totals reconcile. Seventy-eight unpaired hands at sixteen combos each is 1,248, and adding the 78 pairs returns you to 1,326.

Hand	Combos	Chance of being dealt
AA, or any specific pair	6	0.45%
AKs, or any specific suited hand	4	0.30%
AKo, or any specific offsuit hand	12	0.90%
AK, suited and offsuit together	16	1.2%
Any pocket pair at all	78	5.9%

Six combos per pair. Four per suited hand. Twelve per offsuit hand. Sixteen per unpaired hand.

That table explains something most players know by feel without knowing why. You are dealt AK roughly three times as often as AA, and offsuit AK three times as often as suited AK. The 169 squares on a preflop grid look equal and are not, and the offsuit squares carry most of the weight.

The consequence shows up every time you estimate how often somebody holds a class of hand. All thirteen pocket pairs together are 78 combos, while the ten offsuit broadway hands alone are 120. Ranges are dominated by unpaired hands because there are simply more ways to make them, and any read that treats a pair and an unpaired hand as equally likely has already gone wrong.

Every visible card changes the count

Board cards and your own two are cards nobody else can hold. Combos vanish accordingly, and the effect is larger than it feels.

Cards of that rank visible	Pair combos left	AK combos left, aces visible
0	6	16
1	3	12
2	1	8
3	0	4

Pairs halve, then halve again. One card of the rank on the board takes a set from six combos to three; two takes it to a single combo. So "he could have flopped a set" is a much weaker objection than it sounds. On an A♠9♦4♣ flop there are exactly nine set combos in the whole deck — three each of AA, 99 and 44.

Suited hands are the first casualty. A specific suited hand starts with only four combos. On that same A♠9♦4♣ board, A9 suited has two combos left, hearts and clubs, because the spade and diamond versions are already lying face up.

Removal compounds when the board pairs. Two nines on the board leave exactly one combo of 99 in the deck, so the set-or-better portion of everybody's range nearly disappears. That thinning is why paired boards support so much more bluffing than dry unpaired ones: there is much less strength available for anyone to be holding, on either side of the table.

Counting a flop range

Counting is only worth anything applied to a range you believe in, which is what basic hand reading skills are for. Say a tight UTG opener continuation-bets that A♠9♦4♣ flop and you want to know how strong that really is. Restrict them to the hands that genuinely want to build a pot.

Holding	Combos
AA, 99, 44 (sets)	9
AK, AQ (top pair, strong kicker)	24
KK, QQ (overpairs)	12
Total	45

Sets are nine of 45 combos, so one time in five. Top pair is more than half. That is a very different picture from the one your gut supplies when a tight player fires at an ace-high board.

Now give yourself A♥K♥ and count again, with two aces and three kings left unseen.

Holding	Combos, with A♥K♥ in your hand
AA	1
99, 44	6
AK, AQ	14
KK, QQ	9
Total	30

Your two cards deleted a third of his range. Sets went from 20% of it to 23%, because holding an ace makes his ace-high hands less likely while leaving the sets untouched. That sounds like good news until you realise the part you removed is the part you were beating.

The river, where counting pays

You hold A♠Q♦ on Q♠ J♦ 6♣ 3♥ 2♠ and your opponent bets the pot. Top pair top kicker is now a marginal bluff catcher: it beats every bluff and loses to every value hand, so the entire decision collapses into one ratio.

Count his value first, remembering that your own Q♦ removes combos.

• QJ, two pair: two queens left times three jacks, so 6 combos
• QQ: 1 combo
• JJ: 3 combos
• 66: 3 combos

That is thirteen combos of value. A pot-sized bet asks you to be right one time in three, so bluffs need to make up a third of his betting range — at least seven combos against those thirteen.

Then count what could be bluffing. T9 was an open-ender that missed when the 8 never arrived, and KT missed the same straight from the other end. Sixteen combos of each, thirty-two in total, and you block none of them.

He does not need all thirty-two. He needs seven, and he holds thirty-two candidates you have no card in. Call.

Change one card in your hand and the answer moves. Hold K♠Q♦ instead and your king removes four of the sixteen KT combos, so his bluff count drops from thirty-two to twenty-eight against the same thirteen value combos. Still a call, but a thinner one, and the difference came from a single card rather than from anything you read in him.

Blockers are combinatorics under another name

When you hold a card, you delete every combination that needed it. That is the whole mechanism.

The A♠ in your hand on a three-spade board removes every nut-flush combination from an opponent's range in one stroke. The same card does nothing on a board with no spades. Blockers are not a property of your cards, they are a property of your cards against a specific range.

Proportion matters more than the raw number removed. A card that deletes four combos from a twelve-combo calling range has cut that range by a third; the same card against forty combos of overpairs barely registers. Blocker talk goes wrong when players name the card and stop, rather than naming the range and counting what is actually left in it.

The practical version is two questions. Before you bluff, ask what your hand removes from the hands that would call. Before you call, ask what it removes from the hands that would bet. Those questions point in opposite directions, which is why one holding can be an excellent bluff and a poor call, and why how blockers cut combos is worth working through on its own.

Counts worth carrying with you

Situation	Combos
All starting hands	1,326
A pocket pair, nothing visible	6
A pocket pair, one card on the board	3
All sets on an unpaired flop	9
A specific unpaired hand	16
That hand with one of its ranks on the board	12
Top pair with one specific kicker	12
Two pair using two board cards, such as A9 on A-9-x	9

Every published poker statistic is one of these counts divided by 1,326. Being dealt a pocket pair is 78 in 1,326; being dealt aces is 6 in 1,326. The rest of the common poker probabilities are the same arithmetic with more cards taken out of the deck.

Making it automatic

1. Write the range down first. A count applied to a range you invented on the spot is precision without accuracy.
2. Subtract your own cards every single time. This is the step players skip, and it moves the answer by more than any other adjustment.
3. Count value, set the bluff threshold, then hunt for bluffs. Half the value count is the number of bluff combos a pot-sized bet needs to make your call break even.
4. Check yourself after the session. An equity calculator for hand versus range runs the count and the equity together once the range is written out.

None of this needs arithmetic harder than multiplying two small numbers. What it needs is doing it at all. Most players never turn "he might have it" into a number, and that conversion is the entire edge.