Poker combos and combinatorics: counting hands, dead cards and blockers

Why count combos

Saying "they have AA or AK" is not enough: you need to know how often they have each one. Every hand in a range weighs as much as its number of combos, and that difference changes decisions.

AA and AK look like two equal options, but AK has 16 combos and AA only 6. Against that range, 72.7% of the time (16 of 22) you are up against AK. If your hand beats AK and loses to AA, the count tells you calling can be fine even though the hand you fear exists.

Counting combos is the base of three things you will use all the time: equity against a range, the effect of your blockers, and the value-to-bluff mix on the river.

6, 4, 12 and 16: the core numbers

There are 4 cards of each rank, one per suit. From there:

• Pocket pair: 6 combos. You choose 2 of the 4 cards: 4 × 3 / 2 = 6. For QQ: spades-hearts, spades-diamonds, spades-clubs, hearts-diamonds, hearts-clubs, diamonds-clubs.
• Suited hand: 4 combos. One per suit: A♥K♥, A♠K♠, A♦K♦, A♣K♣.
• Offsuit hand: 12 combos. 4 × 4 = 16 ace-king combinations, minus the 4 same-suit ones.
• Unpaired hand in total: 16 combos. 4 suited + 12 offsuit.

In the 13 × 13 grid of the preflop charts each cell is one of those 169 hands, but they do not weigh the same: an offsuit cell is worth three times a suited one and twice a pair.

Dead cards: what is already visible

Every card you can see (on the board or in your hand) is a dead card for your opponent: they cannot hold it. The counts change like this:

• A pair with one card of that rank visible: 3 cards remain, 3 × 2 / 2 = 3 combos. With two visible, 1 combo.
• An unpaired hand: (live cards of the first rank) × (live cards of the second).

Example, flop K♦7♠2♣:

If you also hold A♥Q♣, AK drops to 9 (3 × 3) and AA to 3. Always remove your own cards: they are the ones you know best. What is left are the live combos.

Blockers: your hand changes your opponent's range

A blocker is one of your cards that reduces the combos of certain hands for your opponent. The effect is called card removal.

Preflop example: you 3-bet the CO from the BTN and they 4-bet. Suppose their 4-bet range is QQ+ and AK:

With an ace in your hand you remove 7 combos from your opponent, 20.6% of their range, and precisely from the hands that play best against you. That is why small suited aces are good hands to 3-bet or 4-bet as bluffs: besides playability, they block AA and AK.

Compare with K♥Q♥: the range drops to 24 combos (AA 6, KK 3, QQ 3, AK 12). You block 10 combos, more than with the ace, but none of AA, which is the hand that beats you most.

Counting value vs bluffs on the river

The most profitable use of combinatorics is on the river, with a bluff-catcher. Board K♦9♠5♠2♣3♥: the spade draw missed. The pot is 40 bb and your opponent bets 30 bb (3/4 pot). To call you need 30 / (40 + 60) = 30% equity, meaning at least 30% of their bets must be bluffs.

Suppose they value bet KK, 99, 55 and K9, and bluff with six missed flush draws: AsQs, AsJs, AsTs, QsJs, QsTs and JsTs. Ignoring your hand, value is 3 + 3 + 3 + 9 = 18 combos and bluffs are 6: 25% bluffs. Now count with your hand:

Both hands block value equally (KK drops to 1 and K9 to 6), but K♥Q♠ also blocks three bluffs (AsQs, QsJs and QsTs). The result:

• With K♣8♥: call EV = 6/19 × 70 − 13/19 × 30 = +1.58 bb. You call.
• With K♥Q♠: EV = 3/16 × 70 − 13/16 × 30 = −11.25 bb. You fold.

The hand with the worse kicker is the better bluff-catcher because it does not block your opponent's bluffs. That is the kind of decision that only shows up when you count.

Common mistakes

• Counting hands instead of combos. "They can have AA, KK or AK" is 6 + 6 + 16 = 28 combos, not three equal options.
• Forgetting the board and your hand. With a king on board, KK is no longer 6 combos but 3.
• Picking blockers backwards. To call you want to block their value, not their bluffs; to bluff, the opposite: block the hands that call you.
• Building impossible ranges. If your opponent would never reach the river with AsTs, take it out of the count; combinatorics is only as good as the range you assign.

How to practice it

Start with one fixed drill: on every river where you are unsure, write down 3 or 4 value hands and 3 or 4 bluffs for your opponent and count combos with your hand included. In POKER AULA the Calculator counts combos for you with the board and dead cards, the preflop charts show 6-max cash ranges on the 169-hand grid, and the Training mode puts you in bluff-catcher river spots with the reasoning behind each answer. Combos are covered in the Medium level and blockers in Upper Intermediate. You can try it free for 48 hours.