Combinatorics in poker - how to get the most information from the cards

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Poker is not only a game of chance, but also a fascinating exploration of strategies and tactics.

One of the key aspects of a successful game is the use of combinatorics, the science of combinations and permutations, which helps to make informed decisions in a variety of situations. Let's dive deeper into the world of combinatorics in the context of poker and consider how this tool can be useful for beginners and amateurs in this exciting game.

1. What is combinatorics in poker

Combinatorics is the calculation of the number of possible ways to make a poker combination with a certain set of visible cards.

For example, combinatorics allows you to answer the following questions:

1. In how many ways can pocket aces (AA) be made before the cards are dealt;
2. How many variants of straight draw can be on the J53 flop;
3. What is the number of possible ways to make two pair on the board 972.

In other words, combinatorics gives a quantitative assessment of the probability of certain hands in rivals. This is extremely valuable information that allows you to make optimal decisions in poker.

2. Basic principles of combinatorics

To effectively use combinatorics in the game, you need to know a few basic principles:

Formula: Unpaired Hand Combinations = Number of Free Rank 1 Cards * Number of Free Rank 2 Cards

Here is an example calculation using the formula. Let's say we need to calculate how often an opponent can have AK on the K72 flop.
Solution: Free cards of rank king (K) - 3 pieces (there are 3 cards left in the deck, except for the King on the flop) Free cards of rank Ace (A) - 4 pieces
Substitute into the formula: Combinations AK = 3 * 4 = 12
Answer: On this K72 flop, the enemy can have a maximum of 12 AK combinations.

To count pair combinations, the formula is used: (Available cards of the desired rank) x (Available cards of the desired rank - 1) / 2

Here is an example of counting pair hand combinations using the formula: Let's say you want to know how many QQ combinations an opponent can have on the Q72 flop.
Solution: Available cards of the desired rank - 3 (3 Queens left in the deck, except for the Queens on the flop) Available cards of the desired rank – 1:  (3 – 1) = 2
Substitute into the formula: Combinations QQ = (3) x (2) / 2 = 3
Answer: On this flop, the opponent can have only 3 combinations of pocket QQs. Blocking cards can reduce the number of combinations (on the table or with the player).

Accounting for blockers when counting

Blocking cards are a very important factor affecting combinatorics in poker. Blockers are used on both preflop and postflop. On preflop, only your two pocket cards can be the only blockers. On the post-flop, in addition to your cards, the cards on the table also become blockers. They also reduce the number of combinations in the opponent.

• For example: in preflop, the maximum combination of AA is 6. But if we have A5, then our suit is blocked for opponents. This means that there are a maximum of 3 AA combinations for them.

 For pocket pair, the “rule 6, 3, 1, 0” applies:

Determine the combinations of straight draw on the flop. To calculate the combinations of unpaired hands, you need to multiply the number of free cards of the desired ranks.

Using the familiar formula: Unpaired hand combinations = Number of free Rank 1 cards * Number of free Rank 2 cards

Example: flop
Possible straight draw: J-9 for K-Q-J-T-9, A-J for A-K-Q-J-T
Count: For J-9: J - 4 combinations , 9 - 4 combinations. Combinations J-9 = 4 * 4 = 16 For A-J: A - 4 combinations, J - 4 combinations. Combinations A-J = 4 * 4 = 16
Total straight draw combinations = 16 + 16 = 32

When calculating, take into account the position of the opponent. In this example, the player could attack from an early position, which reduces the likelihood of having J9o in his hand, and this affects the number of final combinations.

3. Combinatorial calculation - a powerful tool for analyzing ranges in poker

Knowing the exact number of possible combinations of different hands allows a much better understanding of the range of opponents. If you simply operate with approximate ranges, without taking into account combinatorics, you risk missing a lot of important nuances. Of course, it is unrealistic to count all the combos on the fly at the table. But even a simple acquaintance with the basics of combinatorics will change your perception of the distribution of hand strength. 

Over time, you will begin to understand that straight draw is more common than it seems at first glance. But dangerous flush draw on the board is usually less than many players fear. Similar insights based on knowledge of combinatorics will help you make more accurate decisions in similar situations in the future. Wrong fears will disappear, giving way to a cold poker calculation. So try to immerse yourself as deeply as possible in the study of combinatorics. This investment of time will pay off handsomely due to a clearer understanding of the game and the right decisions at the poker table.

4. Example of calculations taking into account combinatorics

Key points

So, here are the key points about using combinatorics in poker:

Master combinatorics - and you will significantly increase your edge in poker! 

5. Homework

Input data: on flop
Question: How much maximum can an opponent have with combinations of the following hands:

My main task as a coach is to teach you to think about the hands yourself, so that during the game you can apply the knowledge yourself. Therefore, you can send me answers in private messages to Telegram @ivang212.