• How Many Possible Starting Hands In Texas Hold'em
• How Many Starting Hands In Texas Holdem
• How Many Starting Hands Are There In Texas Holdem

Hand Combinations -- The Basics

1. There are 169 different two card starting hand combinations in hold’em poker. This number assumes, for the sake of argument, that is the same as, or any other suited combination. If you are not dealt a pair, then your starting hand will either be suited or unsuited, and either connected or unconnected (gapped).
2. A pair of queens, also known as 'ladies,' rounds out the top three best starting hands for Texas Hold 'em poker. You will hear many groans from players over this hand. It looks so pretty and it is strong, but they have often had it busted in the past. If an ace or king comes on the flop, you are probably going to be bested.

There are 1326 distinct possible combinations of two hole cards from a standard 52-card deck in hold 'em, but since suits have no relative value in this poker variant, many of these hands are identical in value before the flop.

Hand combinatorics isn't as scary as it sounds.

It's not some big, complex math technique that only a select few guys with PhDs can understand.

It's an easy, highly useful technique that we poker players can use to help put our opponents on hand ranges.

The technique works kind of like it sounds -- we want to calculate the number of combinations of hands our opponent can be holding in a given scenario.

Let's walk through the process of calculating combinations step-by-step. By the time you're done reading this article, you'll be able to more accurately assess what an opponent's hand range could be.

Combinatorics 101: What's a Combination?

A combination is simply a way to put a set of items together where all the items are drawn from a larger set.

In simple English, a combination is a way to pick some stuff out of a bigger pool of stuff.

Say we have a set of objects A, where A = {1, 2, 3, 4, 5}. We want to figure out the number of 2-object subsets we can form out of A; that is, the number of combinations of 2-object subsets we can pick from the set of objects A.

Calculating this turns out to be pretty simple. All we need to do is follow this simple formula:

n = total # of objects in the set
x = number of items we want to choose
C = combinations of x in set n

C = n! / x! * (n-x)!

Plugging in our numbers for set A into this equation, we get:

n = 5
x = 2

C = 5! / 2! * (5-2)! = 10

So there are 10 unique combinations of 2 items that we might choose from set A.

Now, let's say we have a set of 52 items that we want to draw from. As a basic set of 52 objects, B can be the deck of 52 cards for example.

We just plug the total number of objects in the set (52) into our magic equation, along with the number of items we want to include in each chosen subset (2), and get:
n = 52
x = 2
C = 52! / 2! * (52-2)! = 1326

So there are 1,326 unique combinations of 2 items that we might choose from set B.

You will probably find these numbers to be familiar. If so, good! It's probably because 1,326 is the number of possible starting hands in Holdem poker. What we were acually determining in the above calculation was how many combinations of 2 cards we can pick from a 52-card deck.

Keeping with poker, let's examine a more applicable form of calculating combinations, and some situations in which such a method would be useful.

Combinatorics 101: Combinations of Poker Hands

There's obviously no way you'll be able to whip out a calculator at the poker table and calculate combinations. Luckily, there are some more compact ways of calculating combos that don't require calculators or intense math. We can use these benchmarks to help refine our estimates of opponents' ranges.

There are three basic numbers you need to know:

• There are 12 combinations of any given offsuit unpaired hand in Holdem.
• There are 6 combinations of any given paired hand in Holdem.
• There are 4 combination of any given suited unpaired hand in Holdem.

How we arrive at these numbers is pretty basic:

• There are 4 of any card of a given suit in a deck, and 3 of another particular card of a different suit. So we multiply 4 by 3 to obtain 12, the number of combos for a given offsuit unpaired hand.
• There are 4 of any card of a given suit in a deck, and 3 additional cards left in the deck of that rank. So we multiply 4 by 3 to obtain 12. When counting pairs, there are going to be two ways to make each unique combination of pair; for example, 6c6s and 6s6c. So we must divide 12 by 2 to eliminate double-counts. So 12/2 = 6, the number of combinations of a given pair preflop.
• There are 4 of any card of a given suit in a deck, and only one card after that that can make the suited hand. So (4)(1) = 4, the number of combinations for a given suited hand.

A Practical Example

Say you're playing against an opponent whose range you estimate to be {JJ+, AKo, AQs+} in a given situation. You want to break down the number of hand combinations in his range. You'll calculate the combinations as such:

So there are a total of 44 hand combos in your opponent's range, 24 of which are pocket pairs and 20 of which are unpaired hands.

His range looks a lot scarier when we view it as 4 pocket pair hands and 3 unpaired hands, doesn't it? In reality, it's 50/50 that he's got matching cards in the hole; which can change our equity, and thus our correct action, drastically.

One of the top places to play online poker is pokerstars.

For a great training video on poker combinatorics, check out this poker combos video.

'Combinatorics' is a big word for something that isn’t all that difficult to understand. In this article, I will go through the basics of working out hand combinations or 'combos' in poker and give a few examples to help show you why it is useful.

Oh, and as you’ve probably noticed, 'combinatorics', 'hand combinations' and 'combos' refer to the same thing in poker. Don’t get confused if I use them interchangeably, which I probably will.

What is poker combinatorics?

How Many Possible Starting Hands In Texas Hold'em

Poker combinatorics involves working out how many different combinations of a hand exists in a certain situation.

For example:

• How many ways can you be dealt AK?
• How many ways can you be dealt 66?
• How combinations of T9 are there on a flop of T32?
• How many straight draw combinations are there on a flop of AT7?

Using combinatorics, you will be able to quickly work these numbers out and use them to help you make better decisions based on the probability of certain hands showing up.

Poker starting hand combinations basics.

• Any two (e.g. AK or T5) = 16 combinations
• Pairs (e.g. AA or TT) = 6 combinations

If you were take a hand like AK and write down all the possible ways you could be dealt this hand from a deck of cards (e.g. A K, A K, A K etc.), you would find that there are 16 possible combinations.

See all 16 AK hand combinations:

Similarly, if you wrote down all the possible combinations of a pocket pair like JJ (e.g. JJ, JJ, JJ etc.), you would find that there are just 6 possible combinations.

See all 6 JJ pocket pair hand combinations:

So as you can see from these basic starting hand combinations in poker, you’re almost 3 times as likely to be dealt a non-paired hand like AK than a paired hand. That’s pretty interesting in itself, but you can do a lot more than this…

Note: two extra starting hand combinations.

As mentioned above, there are 16 combinations of any two non-paired cards. Therefore, this includes the suited and non-suited combinations.

Here are 2 extra stats that give you the total combinations of any two suited and any two unsuited cards specifically.

• Any two (e.g. AK or 67 suited or unsuited) = 16 combinations
• Any two suited (AKs) = 4 combinations
• Any two unsuited (AKo) = 12 combinations
• Pairs (e.g. AA or TT) = 6 combinations

You won’t use these extra starting hand combinations nearly as much as the first two, but I thought I would include them here for your interest anyway.

It’s easy to work out how there are only 4 suited combinations of any two cards, as there are only 4 suits in the deck. If you then take these 4 suited hands away from the total of 16 'any two' hand combinations (which include both the suited and unsuited hands), you are left with the 12 unsuited hand combinations. Easy.

Fact: There are 1,326 combinations of starting hands in Texas Hold’em in total.

How Many Starting Hands In Texas Holdem

Working out hand combinations using 'known' cards.

Let’s say we hold KQ on a flop of KT4 (suits do not matter). How many possible combinations of AK and TT are out there that our opponent could hold?

Unpaired hands (e.g. AK).

How to work out the total number of hand combinations for an unpaired hand like AK, JT, or Q3.

Method: Multiply the numbers of available cards for each of the two cards.
Word equation: (1st card available cards) x (2nd card available cards) = total combinations

Example.

If we hold KQ on a KT4 flop, how many possible combinations of AK are there?

There are 4 Aces and 2 Kings (4 minus the 1 on the flop and minus the 1 in our hand) available in the deck.

C = 8, so there are 8 possible combinations of AK if we hold KQ on a flop of KT4.

Paired hands (e.g. TT).

How to work out the total number of hand combinations for an paired hand like AA, JJ, or 44.

Method: Multiply the number of available cards by the number of available cards minus 1, then divide by two.
Word equation: [(available cards) x (available cards - 1)] / 2 = total combinations

Example.

How many combinations of TT are there on a KT4 flop?

Well, on a flop of KT4 here are 3 Tens left in the deck, so…

C = 3, which means there are 3 possible combinations of TT.

Thoughts on working out hand combinations.

Working out the number of possible combinations of unpaired hands is easy enough; just multiply the two numbers of available cards.

Working out the combinations for paired hands looks awkward at first, but it’s not that tricky when you actually try it out. Just find the number of available cards, take 1 away from that number, multiply those two numbers together then half it.

Note: You’ll also notice that this method works for working out the preflop starting hand combinations mentioned earlier on. For example, if you’re working out the number of AK combinations as a starting hand, there are 4 Aces and 4 Kings available, so 4 x 4 = 16 AK combinations.

Why is combinatorics useful?

Because by working out hand combinations, you can find out more useful information about a player’s range.

For example, let’s say that an opponents 3betting range is roughly 2%. This means that they are only ever 3betting AA, KK and AK. That’s a very tight range indeed.

Now, just looking at this range of hands you might think that whenever this player 3bets, they are more likely to have a big pocket pair. After all, both AA and KK are in his range, compared to the single unpaired hand of AK. So without considering combinatorics for this 2% range, you might think that the probability break-up of each hand looks like this:

• AA = 33%
• KK = 33%
• AK = 33%

How Many Starting Hands Are There In Texas Holdem

…with the two big pairs making up the majority of this 2% 3betting range (roughly 66% in total).

However, let’s look at these hands by comparing the total combinations for each hand:

• AA = 6 combinations (21.5%)
• KK = 6 combinations (21.5%)
• AK = 16 combinations (57%)

So out of 28 possible combinations made up from AA, KK and AK, 16 of them come from AK. This means that when our opponent 3bets, the majority of the time he is holding AK and not a big pocket pair.

Now obviously if you’re holding a hand like 75o this is hardly comforting. However, the point is that it’s useful to realise that the probabilities of certain types of hands in a range will vary. Just because a player either has AA or AK, it doesn’t mean that they’re both equally probable holdings - they will actually be holding AK more often than not.

Analogy: If a fruit bowl contains 100 oranges, 1 apple, 1 pear and 1 grape, there is a decent range of fruit (the 'hands'). However, the the fruits are heavily weighted toward oranges, so there is a greater chance of randomly selecting an orange from the bowl than any of the 3 other possible fruits ('AK' in the example above).

This same method applies when you’re trying to work out the probabilities of a range of possible made hands on the flop by looking at the number of hand combinations. For example, if your opponent could have either a straight draw or a set, which of the two is more likely?

Poker combinatorics example hand.

You have 66 on a board of A J 6 8 2. The pot is $12 and you bet $10. Your opponent moves all in for $60, which means you have to call $50 to win a pot of $82.

You are confident that your opponent either has a set or two pair with an Ace (i.e. AJ, A8, A6 or A2). Don’t worry about how you know this or why you’re in this situation, you just are.

According to pot odds, you need to have at least a 38% chance of having the best hand to call. You can now use combinatorics / hand combinations here to help you decide whether or not to call.

Poker combinatorics example hand solution.

First of all, let’s split our opponent’s hands in to hands you beat and hands you don’t beat, working out the number of hand combinations for each.

Adding them all up…

Seeing as you have the best hand 79% of the time (or 79% 'equity') and the pot odds indicate that you only need to have the best hand 38% of the time, it makes it +EV to call.

So whereas you might have initially thought that the number of hands we beat compared to the number of hands we didn’t beat was close to 50/50 (making it likely -EV to call), after looking at the hand combinations we can see that it is actually much closer to 80/20, making calling a profitable play.

Being able to assign a range to your opponent is good, but understanding the different likelihoods of the hands within that range is better.

Poker combinatorics conclusion.

Working out hand combinations in poker is simple:

• Unpaired hands: Multiply the number of available cards. (e.g. AK on an AT2 flop = [3 x 4] = 12 AK combinations).
• Paired hands: Find the number of available cards. Take 1 away from that number, multiply those two numbers together and divide by 2. (e.g. TT on a AT2 flop = [3 x 2] / 2 = 3 TT combinations).

By working out hand combinations you can gain a much better understanding about opponent’s hand ranges. If you only ever deal in ranges and ignore hand combinations, you are missing out on useful information.

It’s unrealistic to think that you’re going to work out all these hand combinations on the fly whilst you’re sat at the table. However, a lot of value comes from simply familiarising yourself with the varying probabilities of different types of hands for future reference.

For example, after a while you’ll start to realise that straight draws are a lot more common than you think, and that flush draws are far less common than you think. Insights like these will help you when you’re faced with similar decisions in the future.

The next time you’re doing some post session analysis, spend some time thinking about combinatorics and noting down what you find.

Poker combinatorics further reading.

Hand combinations in poker all stem from statistics. So if you’re interested in finding out more about the math side of things, here are a few links that I found helpful:

• Combinations video - Youtube (all the stuff on this channel is awesome)

If you’re more interested in finding out more about combinations in poker only, here are a few interesting reads:

Go back to the awesome Texas Hold'em Strategy.

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