Poker Combinatorics

Poker Combinatorics Calculator

Combinations in Poker

Order generally doesn't matter in poker. So counting hands means counting combinations. Choosing r cards from n possible choices: nCr = n! / (r!(n − r)!). There are 1,326 combinations of Hold'Em starting hands — 52 choose 2 — and 270,725 combinations of Omaha starting hands — 52 choose 4.

Those 1,326 Hold'Em combinations can be grouped into pocket pairs, suited hands and offsuit hands. There are 13 groups of 6 pocket pairs, 78 groups of 4 suited hands and 78 groups of 12 offsuit hands: (13 × 6) + (78 × 4) + (78 × 12) = 1,326.

Combinations show why many players think Omaha is harder than Hold'Em. At showdown an Omaha player must use exactly two hole cards and three board cards: 4C2 × 5C3 = 6 × 10 = 60 ways to make a hand, compared with 7C5 = 21 in Hold'Em.

The Factorial Function

The "!" means factorial: multiply a number by every whole number below it, all the way down to 1. For example, 5!, pronounced "5-factorial", is 5 × 4 × 3 × 2 × 1 = 120, and by definition 0! = 1.

Factorials grow fast: 52! is a 68-digit number. Luckily, most of it cancels out. For 52 choose 2:

Permutations are when order matters: nPr = n! / (n − r)!. The difference is the r term: r! is dropped from the denominator.

That's exactly 2! = 2 times the 1,326 combinations, because A♠K♥ and K♥A♠ now count as two different orders. In general there are always r! times as many permutations as combinations. In Hold'Em, after the flop, there are 47P2 = 2,162 ordered turn-and-river pairs, but only 47C2 = 1,081 runouts.

Counting Combinations with Blockers

In Hold'Em, an unpaired hand like AK chooses 1 of the 4 aces and 1 of the 4 kings: 4C1 × 4C1 = 4 × 4 = 16 combinations: 4 suited and 12 offsuit. A pocket pair like QQ chooses 2 of the same 4 queens: 4C2 = (4 × 3) / 2 = 6 combinations.

In Omaha, AAKK has 4C2 × 4C2 = 36 combinations, and only 6 of them are double suited.

Known cards remove combinations from your opponent's possible hands. Remove an ace and the combinations of AK drop from 16 to 12; remove the A♥ and K♥ and they drop to 9.

Hold'Em FAQ

How many starting hands are there in Texas Hold'Em?

There are 1,326 combinations of starting hands in Texas Hold'Em. A Hold'Em hand is 2 cards from a 52-card deck: 52C2 = (52 × 51) / (2 × 1) = 1,326. Grouped by rank and suitedness they make 169 distinct starting hands: 13 pocket pairs, 78 suited hands and 78 offsuit hands.

See all 169 hands on the Hold'Em chart ↑

What is the best starting hand in Texas Hold'Em?

Pocket aces are the best starting hand in Texas Hold'Em. Against a random hand, AA has 85.19% equity in our simulations (1,000,000 trials per combination), followed by KK (82.42%) and QQ (79.91%). AK, often called the best hand that isn't a pair, ranks 8th suited (67.05%) and 12th offsuit (65.33%). In actual play, AA is still one pair, so it is at its strongest heads-up and loses value as more players see the flop.

What are the odds of pocket aces winning?

Pocket aces win about 85% of the time against a random hand: 84.92% outright, plus a 0.54% chance of a split pot, for 85.19% equity in our simulations. Against pocket kings, the next-best hand, AA has 81.95% equity over every possible board, about 4.5 to 1.

What is equity in poker?

Equity is your share of the pot if the hand were played out many times: the hands you win, plus half of the hands you tie. Our rankings use equity against a random hand. 3-2 offsuit, for example, wins 29.24% of the time and splits the pot 6.13% of the time, so its equity is 29.24% + 6.13% / 2 ≈ 32.30%.

How many combos of AK are there?

There are 16 combinations of AK. AK takes 1 of the 4 aces and 1 of the 4 kings: 4C1 × 4C1 = 16. 4 are suited (AKs) and 12 are offsuit (AKo). Every unpaired hand has the same 16 combinations.

How many combos of a pocket pair are there?

There are 6 combinations of a pocket pair. A pair like QQ takes 2 of the 4 queens: 4C2 = (4 × 3) / 2 = 6.

What is the worst hand in poker?

By the numbers, 3-2 offsuit is the worst hand in poker. Against a random hand it has 32.30% equity, the lowest of all 169 starting hands; 7-2 offsuit has 34.58% and ranks 165th, ahead of 5-2, 6-2, 4-2 and 3-2 offsuit. In actual play, many players still consider 7-2 offsuit the worst: 3-2 can make a straight, including the wheel (A-2-3-4-5), while 7-2 can't make a straight using both cards, so it rarely wins a big pot.

Why is A8 called the Dead Man's Hand?

Legend says Wild Bill Hickok was holding two pair, aces and eights, when he was shot at a poker table in Deadwood, Dakota Territory, in 1876. As a Hold'Em hand, A8 offsuit has 59.88% equity against a random hand, but only 26.07–28.29% against AK, AQ, AJ or AT, so it gets worse the more players there are in the hand.

The Dead Man's Hand: the legend, the odds and the math →

What are blockers in poker?

Blockers are cards you can see, in your hand or on the board, that your opponent therefore can't hold. They remove combinations from your opponent's possible hands, which is why the idea is also called card removal. Hold an ace and AK drops from 16 combinations to 12, and AA drops from 6 to 3. Hold A♥K♥ and AK drops to 9.

What are the odds of being dealt a pocket pair?

The odds of being dealt a pocket pair are about 1 in 17. There are 13 pairs with 6 combinations each: 78 / 1,326 = 5.88%.

What are the odds of being dealt pocket aces?

The odds of being dealt pocket aces are 1 in 221. AA has 6 combinations: 6 / 1,326 = 0.45%. The same is true of any specific pair.

What are the odds of being dealt suited cards?

The odds of being dealt suited cards are about 1 in 4. There are 312 suited combinations: 312 / 1,326 = 23.53%.

How many possible flops are there in Hold'Em?

There are 19,600 possible flops once you hold your two cards: 50C3 = 19,600. From a full deck there are 52C3 = 22,100.

How many possible boards are there in Hold'Em?

There are 2,118,760 possible five-card boards once you hold your two cards: 50C5 = 2,118,760.

Omaha and PLO FAQ

How many starting hands are there in Omaha (PLO)?

There are 270,725 combinations of starting hands in Omaha. An Omaha hand is 4 cards from a 52-card deck: 52C4 = (52 × 51 × 50 × 49) / (4 × 3 × 2 × 1) = 270,725.

How many distinct starting hands are there in Omaha?

There are 16,432 distinct starting hands in Omaha, counting hands that differ only by suit once: A♠A♥K♠K♥ plays the same as A♦A♣K♦K♣. In Hold'Em the same grouping gives 169.

See every Omaha hand in the tesseract ↑

What is the best starting hand in Omaha (PLO)?

Against one random hand, AATT double suited is the best starting hand in Omaha, with 71.54% equity in our simulations (1,000,000 trials per combination), ahead of AAJJ (71.22%) and AAQQ (70.92%), all double suited. AAKK, often called the best hand, has 70.67%: no better than AA55 (70.77%) or AA99 (70.73%) double suited. We hypothesize the reason is straights. Every straight contains a 5 or a 10, and holding two of the middle cards blocks the straights an opponent needs: opponents beat AAKK with a straight 10.05% of the time, but AATT, AA55 and AA99 only 8.47–8.65%. In multiway pots, many players prefer AAJT double suited, whose four cards all work together; heads-up it has 70.62% equity and ranks in the top ten.

What is the worst starting hand in Omaha (PLO)?

Four deuces (2222) is the worst starting hand in Omaha, with only 9.31% equity against a random hand, followed by 3333 (12.19%), 4444 (15.43%) and 5555 (18.86%). Because you must play exactly two hole cards, every pair you can use is a pair of deuces: no straights, no flushes, and no set, since the other two deuces are in your own hand. After the low quads come trip deuces with one side card, like 7222 (20.79%).

How many two-card combinations are in an Omaha hand?

Every Omaha hand contains 6 two-card combinations: 4C2 = 6. You must use exactly 2 of your 4 hole cards with 3 of the 5 board cards, so at showdown you have 4C2 × 5C3 = 6 × 10 = 60 possible five-card hands.

Break down any Omaha hand ↑

What does double suited mean in Omaha?

A double-suited hand has two cards of one suit and two of another, like A♠K♠Q♥J♥. 36,504 hands are double suited, 13.48% of all Omaha hands. Single suited (2-1-1) is 58.43%, unsuited or rainbow (1-1-1-1) is 10.55%, three of one suit is 16.48% and all four of one suit is 1.06%.

What are the odds of being dealt aces in Omaha?

The odds of being dealt aces in Omaha are about 1 in 39. 6,961 Omaha hands hold at least two aces: 6,961 / 270,725 = 2.57%. In Hold'Em, AA is 1 in 221.

How many combinations of AAKK are there?

There are 36 combinations of AAKK: 4C2 × 4C2 = 6 × 6 = 36. 6 are double suited, 24 single suited and 6 unsuited. Double-suited AAKK is dealt 1 in 45,121 hands.

How often are you dealt a pair in Omaha?

You are dealt at least one pair in about 1 in 3 Omaha hands. 87,685 Omaha hands hold at least one pair: 87,685 / 270,725 = 32.39%.

How many possible flops are there in Omaha?

There are 17,296 possible flops once you hold your four cards: 48C3 = 17,296. There are 48C5 = 1,712,304 possible five-card boards.

Combinatorics FAQ

What is combinatorics in poker?

Poker combinatorics is counting how many ways a hand or board can be dealt. Players use it to count an opponent's possible hands, weigh one holding against another, and see how the cards they hold change those counts.

What are combos in poker?

Combos is poker shorthand for combinations: the specific ways a hand can be dealt. AK has 16 combos (4 suited and 12 offsuit), a pocket pair has 6, and counting an opponent's combos is how players weigh what they might hold.

How do you calculate combinations (nCr)?

Combinations are calculated with nCr = n! / (r! × (n − r)!). Multiply r numbers counting down from n, then divide by r!. For a Hold'Em hand: 52C2 = (52 × 51) / (2 × 1) = 1,326.

What is the difference between nCr and nPr?

nCr counts selections where order does not matter, like a poker hand: n! / (r! × (n − r)!). nPr counts arrangements where order matters: n! / (n − r)!. There are always r! times as many permutations as combinations.

How many 5-card poker hands are there?

There are 2,598,960 five-card poker hands: 52C5 = 2,598,960.