Poker Hands Permutations Combinations
The answer is 2598960 combinations or in other words, this hand occurs on average once in 2 million 598 hundred thousand 960 hands! Just as likely is a hand with the 8 of hearts, 3, 4 and jack of clubs, and 9 of spades (but you would not likely win at poker with this latter hand). Permutations vs. Combinations Both are ways to count the possibilities The difference between them is whether order matters or not Consider a poker hand: A, 5, 7♣, 10♠, K♠ Is that the same hand as: K♠, 10♠, 7♣, 5, A Does the order the cards are handed out matter? If yes, then we are dealing with permutations. Once a hand is highlighted, the number of combinations of that particular hand is displayed in the same box. The number of total hand combinations in the range is displayed under the slider bar as both a fraction and a percentage. In this case, the range contains 74 combinations, or 5.58% of all hands. Take Your Poker Game to the Next Level.
Poker in 2018 is as competitive as it has ever been. Long gone are the days of being able to print money playing a basic ABC strategy.
Today your average winning poker player has many tricks in their bags and tools in their arsenals. Imagine a soldier going into the heat of battle. Without his weapons, he is practically useless, and chances of survival are extremely low.
If you sit down at a poker table without any preparation or general understanding of poker fundamentals, the sharks are going to eat you alive. Sure you may get lucky once in a blue moon, but over the long term, things won’t end well.
With the evolution of poker strategy, you now have many tools at your disposal. Whether it be online poker training sites, free YouTube content, poker coaching, or poker vlogs, there’s no excuse to be a fish in today's game.
Some of the essential fundamentals you need to be utilizing that every poker player should have in their bag of tricks whether you are a Tournament or Cash Game Player are concepts such as hand combinations (Also known as hand combinatorics or hand combos).
Hand Combinations and Hand Reading
If you were to analyze a large sample of successful poker players you would notice that they all have one skill set in common: Hand Reading
What does hand reading have to do with hand combinations you might ask?
Well, poker is a game of deduction and to be a good hand reader, you need to be good at correctly ranging your opponents.
Once you have assigned them a range, you will then need to start narrowing that range down. Combinatorics is one of the ways we do this.
So what is combinatorics? It may sound like rocket science and it is definitely a bit more complex than some other poker concepts, but once you get the hang of combinatorics it will take your game to the next level.
Combinatorics is essentially understanding how many combos each of your opponent's potential holdings are and deducing their potential holdings utilizing concepts such as removal and blockers.
There are 52 cards in a deck, 13 of each suit, and 4 of each rank with 1326 poker hands in total. To simplify things just focus on memorizing all of the potential combos to start:
- 16 possible hand combinations of every unpaired hand
- 12 combinations of every unpaired offsuit hand
- 4 combinations of each suited hand
- 6 possible combinations of pocket pairs
Here is a short video example of using combinatorics to count the number of ways a non-paired hand AK can be arranged (i.e. how many combos there are):
So now that we have this memorized, let's look at a hand example and how we can apply combinatorics in game.
We hold A♣Q♣ in the SB and 3bet the BTN’s open to 10bb with 100bb stacks. He flats and we go heads up to a flop of
A♠ 5♦ 4♦
We check and our opponent checks back with 21bb in the middle
Turn is the 4♥
We bet 10bb and our opponent calls for a total pot of 41bb
The river brings the 9♠
So the final board reads
A♠ 5♦ 4♦ 4♥ 9♠
We bet 21bb and our opponent jams all in leaving us with 59bb to call into a pot of 162bb resulting in needing at least 36% pot equity to win.
Our opponent is representing a polarized range here. He is either nutted or representing missed draws so we find ourself in a tough spot. This is where utilizing combinatorics to deduce his value hands vs bluffs come into play. Now we need to narrow down his range given our line and his line. Let's take a look at how we do this...
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Blockers and Card Removal Effects
First, let's take a look at the hands we BLOCK and DON’T BLOCK
Since we hold an Ace in our hand and there is an Ace on the board, that only leaves 2 Ace’s left in the deck. So there is exactly 1 combo of AA.
We BLOCK most of the Aces he can be holding, so we can REMOVE some Aces from his range.
We do not BLOCK the A♦ as we hold A♣Q♣, and the A on the board is a spade, so it is still possible for him to have some A♦x♦ hands.
We checked flop to add strength to our check call range (although a bet with a plan to triple barrel is equally valid in this situation SB vs BTN) and because of this our opponent may not put us on an A here.
If he is a thinking player his jam can exploit our thin value bet on the river turning his missed straight/flush draws into a bluff to get us to fold our big pocket pairs and even make it a tough call with our perceived weak holdings.
The problem in giving him significant credit for this part of his bluffing range is the question of would he really shove here with good SDV (Showdown Value)?
These are the types of questions we must ask ourselves to further deduce his range along with applying the combinatoric information we now have.
Now, let's look at all the nutted Ax hands our opponent can have.
If he has a nutted hand like A4 or A5, and we assume he is only calling 3bets with Axs type hands, the only suited combo of those hands he can have are exactly A♥5♥. He can’t have A♦5♦ or A♦4♦ because the 4 and the 5 are both diamonds on the board blocks these hands.
Lets take a look at all of this value hands:
There is only 1 combo of 44 left in the deck, 2 combos of A9s, 3 Combos of 55, 3 Combos of 99, 2 Combos of 45s - some of these hands may also be bet on the flop when facing a check.
So to recap we have:
1 Combo A5s, 2 Combos of A9s, 3 Combos of 55 (With one 5 on board, the number of combinations of 55 are cut in half from 6 combos to 3 combos), 1 Combo of 44, 2 Combos of 45s, 3 Combos of 99
Total: 12 Value Combos
Now we need to look at our opponent's potential bluffs
Based on the villain's image, this is the range of bluffs we assigned him:
K♦Q♦(1 Combo), J♦T♦ (1 Combo), T♦9♦ (1 Combo), 67s (4 Combos)
He may also turn some other random hands with little showdown value into bluffs such as A♦2♦/A♦3♦
Total: 9 Bluff Combos
9(Bluff Combos) + 12(Value Combos) = 22
9/21 = 42% of the time our opponent will be bluffing (assuming he always bets this entire range)
11/21 = 58% of the time our opponent will be value raising
Now, this is the range we assigned him in game based on the action and what we perceived our opponents range to be.
We are not always correct in applying the exact range of his potential holdings, but so long as you are in the ballpark of that range you can still make quite a few deductions to put yourself in the position to make the correct final decision.
According to the range we assigned him, he has 11 Value Combos and 9 Bluff Combos which gives us equity of 42%. This would result in a positive expected value call as we only need 36% pot odds to call.
However, unless you are playing against very tough opponents you will not see someone bluffing all 9 combos we have assigned - most likely they will bluff in the range of 4-6 combos on average which gives equity in the range of 20-30% equity. This is not enough to call.
We ultimately made our decision based on the fact that we felt our opponent was much less likely to jam with his bluffs in this spot. Given that it was already a close decision to begin with, we managed to find what ended up being the correct fold.
Now this all may seem a bit overwhelming, but if you just start taking an extra minute on your big decisions you’d be surprised how quickly you can actually process all this information on this spot.
A good starting point is to simply memorize all of the possible hand combinations listed above near the beginning of the article.
Get access to our 30-minute lesson on Combinatorics and PokerStove by clicking on one of the buttons below:
Conclusion On Combinatorics
Eventually accounting for your opponent's combos in a hand will become second nature. To get to the point that , a lot of the work needs to be done off the table and in the lab. As you spend more time studying it and reviewing hand histories like the one above, you will find yourself intuitively and almost subconsciously using combinatorics in your decision making tree.
Poker Hands Permutations Combinations Calculator
But the work will be worth the effort, as being able to count combos on the fly will add a new dimension to your game, allow you to make more educated decisions, become a tougher opponent to play against and move away from playing ABC poker.
Want more content like the ones in this blog post on poker combinatorics? Check out our Road to Success Course where we have almost 100 videos like this to help take your game to the next level. You can also get the first module of the Road To Success Course for Free - for more details see the free poker training videos page by TopPokerValue.com.