Hand Combinations -- The Basics

• 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

This actually makes Omaha much tougher to crack than Hold'em. With two hole cards, you have 1,326 possible starting hands, including duplicates like Ace-Seven of Hearts and Ace-Seven of Diamonds. With four hole cards, you have 270,725 possible starting hands and many more different ways to connect with the board. If you notice how many starting hands are included in the lists above, you’ll see that you have 66 playable hands in these admittedly somewhat arbitrary categories. You have 169 possible starting hands in Texas holdem, which means that I’m recommending you only play the top 39% of the possible starting hands. There are many variations of poker, with Texas Hold ‘Em being the most popular worldwide. In total, there are 2,598,960 possible poker hands with 52 cards. As you can see, half of all profits are generated by only 5 of the 169 starting hand combinations. And with a total of 14 of the 169 starting hands (6.3% of all hole cards dealt), 80% of all profits are made. So this should make it obvious that you should only play few hole cards in Texas Hold’em Cash Games.

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:

Hand Type	Combos Per Hand	Number of Hands	Total Combos
Pocket Pair	6	4	24
Unpaired Offsuit	12	1	12
Unpaired Suited	4	2	8

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.

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There are over one hundred possible starting hands in Texas Holdem. That’s just if you take unique combinations. Start taking into account suit, and the number increases dramatically.

Statistically speaking, you’re just as likely to get any one of these starting hands as any other, though there are certain hands you’d much rather get than others. Also, in terms of game theory, some opening hands are a lot more interesting than others. You know what you’re going to do with a pair of aces. Do you know what you’re going to do with ace-two though?

Unfortunately, it’s impossible in place to go through the strategy of every opening hand in poker, we can highlight a certain number of more interesting hands. I’ve chosen five hands from the list of possible hands to discuss below in greater detail. You are likely to run into them if you play often enough, and their relative strengths present interesting challenges.

1 – Pair of Twos

Congratulations… You have a pair. Aren’t you lucky?

Sure. Go ahead and start grumbling now because of all the pairs you have. This one is undoubtedly, statistically the worse pair you can have. Yes, it will beat any hand that doesn’t have a pair, but how often are you likely to have any opponents that don’t have anything and stay in the game?

The answer is not that many. Don’t worry, though, starting off with a pair of twos isn’t as bad as you think if one of two things are true:

1. You have position
2. There aren’t many people at the table

According to computer models, you are dead even to win with a pair of twos in straight-up play and you will win about a third of the time in a three-player game. Your odds of winning drop off pretty dramatically from there, however. The only thing that could save you, though, is position.

How Many Possible Starting Hands In Texas Hold'em

In a lot of ways, a pair of twos is like a stealth attack. As your opponents survey the board and try to figure out what you have, they’re probably not thinking you have pocket twos. So, if you draw into a set or even manage to get three, four, five, six on the board, there’s a chance you can sneak up on the others and win some money.

Still, to win with this hand more often than not, you need to know what the other players are going to do. That’s why you need position. If you find that everyone else is betting aggressive, there’s a good chance your twos are no good, so get out of the hand. If, however, everyone is lukewarm, you might have some money to make.

How Many Starting Hands In Texas Holdem

The Strategy for This Hand Is as Follows:

How Many Starting Hands Are There In Texas Holdem

If you’re in a game with any more than three people and you don’t have position, fold and fight another day. If you do have position, call any modest bets or make a medium bet of your own. If someone comes at you hard and raises or re-raises, unless you are working their tells, fold and get them in the next hand.

After the flop and each subsequent card, take the pulse of the table. If someone starts to bid more aggressively and you didn’t get a third two, you shouldn’t be out too much money.

After the last card is dealt and no one feels good about their cards, you have to decide how aggressive you want to be. My advice is to always keep things on the low to moderate side of bidding (think fractions of a pot). Even if someone missed their draw, you might be able to get money from them if you bet lower amounts than getting too aggressive in the end.

2 – Pair of 10s

In some ways, a pair of 10s is going to be like a pair of twos. There are a lot more hands that pocket 10s will beat than pocket twos, but there are a bunch of hands that you’d probably rather have.
Then again, this is poker, and you have to play the hand you were dealt and, surprisingly, pocket 10s are a lot better than you might think. In head-to-head play, pocket 10s win about three-fourths of the time and rarely wins less than 20% of the time, no matter the number of opponents.

On the other hand, pocket 10s don’t beat a lot of hands that people are likely to keep. Also, 10s can appear at the end, middle, or beginning of a straight, so you need to be careful that if you get a set of 10s, that someone else didn’t just complete a straight and are now set to take you down.

Therefore, you really have two options with pocket 10s and how you play them is up to you and your playstyle. If you like to play tight or are light on chips and you have more than two opponents, toss the 10s. You’d do better not taking the risk.

If you’re in the mood to get risky or you have a fewer number of opponents, play 10s like they were much better cards. Be aggressive, make the table think you’re sitting on a gold mine, at least until the flop. If you get your set and there’s no obvious straight draw, keep playing aggressively.

If you don’t get that set, you can still play tough because you do have good cards, and you should be able to scare off the table with them if you bet from a position of strength.

3 – King-Queen Suited

King-queen suited is going to win the hand a little over 60% in head-to-head games, but quickly loses its power in larger games. You have the start off a pretty powerful straight and the chance to make two decent pairs.
If you are dealt king-queen suited to start off, you have to at least make a bet preflop and call just about anything unless someone goes all-in or is just being crazy.

The thing you have to remember is that your opening hand is worthless without some help. You need to be dealt another king or a queen while avoiding an ace to have a winning hand. That’s what you need to call preflop so that you can see what help the board gives you.

Ideally, you want a queen on the flop with no ace. This gives you the best pair with the second best kicker. Either way, feel free to bet aggressively throughout the hand until an ace shows up. Then, you have to assume that someone has a literal ace in the hole unless their betting shows you otherwise.

One last thing, keep an eye out for a flush. That’s one thing this starting hand gives that you can use to your advantage and is another reason to see that flop.

4 – Ace-Two Offsuit

The strategy for ace-two is a lot like the strategy for pocket twos. It’s good at small tables or when you have position.

If neither of those two things are true, you might want to consider folding. Yes, it can feel painful to fold an ace, since it’s an ace after all. However, there are others at the table who probably also have aces in the hole and chances are, they have better kickers than you.

5 – Queen-Jack Offsuit

Percentage-wise, queen-jack suited wins almost the same amount as king-queen suited. It also has a sharp drop off in its efficiency as the number of players climb.

Still, in a lot of ways, this hand should be played about the same way as king-queen suited except that you don’t have the chance to earn a flush draw. You also can’t be quite as aggressive with queen-jack as you can with king-queen because jacks are easier to beat than kings.

Still, preflop, bet with confidence. Feel free to call any reasonable opening bet and don’t be afraid to call a bet outside your comfort zone so that you can see the flop. Also, don’t be afraid to make your own bet preflop, though I wouldn’t bet a significant amount because there are several ways to beat you.

Once you see the flop, you have decisions to make. If you didn’t flop a pair or aren’t set up for a straight, it might be a good time to get out of the hand since your chances of success aren’t as high as king-queen. Then again, if no one is betting aggressively, there’s no harm in riding the hand out and testing your luck.

Conclusion

Hopefully, seeing how these five hands can be played will give you confidence next time you see them. As always, poker is a dynamic game and circumstances can change. The best poker strategy will change from hand to hand, but I think the following strategy will help make you some money if you follow it.

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