tools
Combo counter.
There are six ways to hold a pair, four to hold a suited hand and twelve to hold an offsuit one. Every card you can see removes some of them. This counts what is left, which is the one piece of poker arithmetic with nothing estimated in it at all.
the counter
your hand and the board together, like Ah Kd Qc 7s
AA for a pair, AKs for suited, AKo for offsuit, AK for both
Combinations left
.
Before your cards were removed
.
Blocked away
.
| Hand | Left | Normally | Share of the total |
|---|
the same numbers, spelled out
The example holds an ace, a king and a queen, which is why pocket aces drops from six combinations to three and ace king suited from four to two.
the three numbers to know
Six, four and twelve.
Learn these three and most combination questions answer themselves. They are also why a hand type and a hand are not the same thing: there are as many hand types as squares on the starting hand grid, but the squares are not equally likely.
| Kind of hand | Example | Combinations each | Hand types | All of them |
|---|---|---|---|---|
| A pocket pair, aces for example four suits taken two at a time | 6 | 13 | 78 | |
| Two suited cards, ace king suited one for each suit | 4 | 78 | 312 | |
| Two offsuit cards, ace king offsuit four suits for the first, three left for the second | 12 | 78 | 936 |
counted at build time from all 1326 ways of dealing two cards, and the three rows add up to exactly that
blockers
Every card you can see is a card they cannot have.
Hold and your opponent's pocket aces drop from 6 combinations to 3, and their ace king from 16 to 9. Nothing about their behaviour changed. There are simply fewer cards left with which to hold those hands, and that is a fact rather than a read.
| What you can see | Pocket aces | Ace king, any | Ace king suited |
|---|---|---|---|
| Nothing removed | 6 | 16 | 4 |
| You hold one ace | 3 | 12 | 3 |
| You hold two aces | 1 | 8 | 2 |
| One ace in your hand, one on the board | 1 | 8 | 2 |
| You hold an ace and a king | 3 | 9 | 3 |
what combinations are for
Counting value against bluffs.
The practical use of counting combinations is checking whether your betting range is balanced. Your bet size sets the share of bluffs that makes an opponent indifferent to calling, and counting tells you whether your actual hands come anywhere near it. The last column makes it concrete: if you have twelve combinations of value hands, that is how many bluffs belong beside them.
| Bet size | Share that should be bluffs | Value per bluff | Bluffs beside 12 value combos |
|---|---|---|---|
| A third of the pot | 20.0% | 4.00 to 1 | 3.0 |
| Half the pot | 25.0% | 3.00 to 1 | 4.0 |
| Two thirds of the pot | 28.6% | 2.50 to 1 | 4.8 |
| The size of the pot | 33.3% | 2.00 to 1 | 6.0 |
| Twice the pot | 40.0% | 1.50 to 1 | 8.0 |
Read the first row and the last together. A small bet lets you bluff rarely; a large one obliges you to bluff often, and if you do not have the hands for it, the size is wrong rather than the plan. This is the same break even arithmetic as the expected value of a bluff, seen from the counting side instead of the money side.
using it honestly
Combinatorics gives you a ceiling, not a read. It tells you how many ways a hand can exist, and it says nothing about whether this player would have taken this line with it. The useful order is to count first and then subtract what their betting rules out, rather than to start from a suspicion and hunt for numbers that agree with it. The ranges people actually open from each seat are on the range charts, and they are a better starting point than a guess.
The counts also feed straight into equity. How often a hand wins depends on which combinations of it remain, which is why the probability tables count every flop rather than sampling them, and why the hand odds page counts every five and seven card combination. It is all the same subtraction at different scales.
One caution about blockers in particular. They are real and they are small. Holding an ace cuts your opponent's aces by half, and that matters at the margin of a close decision, not in place of one. A blocker is a tiebreaker, and treating it as a reason is how people talk themselves into calls that the price never justified.
next
Counting only helps if you know what to count.
A combination count is only as good as the range you point it at. The charts show which hands come from which seat, which turns this calculator from arithmetic into a read.
questions about combinations
How many combinations of each starting hand are there?
Six for any pocket pair, four for a suited hand and twelve for an offsuit hand. Across all 169 hand types that is 78 paired combinations, 312 suited and 936 offsuit, which adds up to the 1326 ways of dealing two cards from 52.
Why does ace king have sixteen combinations?
Four aces times four kings is sixteen. Four of those sixteen have both cards in the same suit, which leaves twelve offsuit. That is why an unpaired hand comes up more than twice as often as a pair: sixteen ways against six.
What is a blocker?
A card in your hand or on the board that your opponent cannot also have. Holding one ace cuts the ways they can hold pocket aces from 6 to 3, and every ace visible anywhere cuts it again. It is the most reliable read in poker because it is not a read at all, it is subtraction.
How do I count combinations with a board out?
The same way, minus the cards you can see. Every card on the board and in your hand is gone from the deck, so a hand needing that card has fewer ways to exist. Enter your cards and the board in the calculator on this page and it does the subtraction for you.
How many bluffs should I have?
That follows from your bet size, and it is a division rather than a feeling. Betting the pot means your opponent needs 33.3 percent equity to call, so 33.3 percent of the hands you bet should be bluffs: two value hands for every bluff. A half pot bet needs a quarter bluffs, three value hands per bluff. Counting combinations is how you check whether your actual range matches.
Does combinatorics work against real opponents?
It gives you the ceiling, not the answer. Combinatorics tells you how many ways a hand can exist; it says nothing about whether this particular player would have played it this way. The right use is to start from the count and then remove what their line rules out, rather than starting from a feeling and looking for numbers to support it.
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