Explanation of Combination

The term “combinations” (combos for short) is mostly used when discussing either Hold’em or Omaha variants, although it can presumably be used to describe a specific combination of cards in any variant.

There are 1326 unique combinations of two cards that can be dealt preflop in Hold’em. This includes cards which are otherwise identical aside from the suits. I.e both 78♣ and 78♦ are different combinations of hole-cards despite the fact that they are identical in strength. If we ignore any suits, there are actually 169 unique holdings that can be dealt preflop in Hold’em.

There a 4 combinations of each suited hand, e.g. 78
There are 12 combinations of each unsuited hand (not including pocket pairs), e.g. AQ♦.
There are 6 combinations of each pocket pair, e.g. 66

Of course, the possible combinations of each starting hand change after the community cards are dealt.  For example, on a KT6 flop there are no longer 6 combinations of TT that may appear as a player’s hole cards. (More on counting combinations in the strategy application section).

In a four-card variant of poker such as Pot Limit Omaha, the number of possible starting hand combinations increases exponentially. There are 270,725 possible combinations of four-card starting hands in PLO. 

Example of Combination used in a sentence -> (Hold’em) On a K72 board there are 3 combinations of each possible set for a total of 9 set combos.

How to Use Combination as Part of Your Poker Strategy

Here we will discuss how to count combinations in a postflop Hold’em scenario. Before we do so, let’s answer a relevant question. Why would we want to count combinations in Hold’em?  By considering the precise number of certain types of holding within our opponent’s range we can gain insights into what the best play might be. Employing combinatorics usually implies we are in a close situation. If our decision was easy, then counting our opponent’s combinations would be unnecessary.  Consider the following example -

Our opponent makes a half pot bet on the river in a situation where we just hold a bluffcatcher. We calculate that he has 15 legitimate value combinations. How many bluff combinations does he need to show up with in order for us to have a profitable bluff-catch?

To answer this question, we firstly need to calculate our pot-odds. If we made the call we would be investing 25% of the total pot, meaning we need to win more than 25% of the time in order to make profit. (Also expressed as 3:1 pot odds). We hence need our opponent to be bluffing at least 25% of the time in order for our call to break even. If our opponent hence had 5 bluff combos our call would break even. If he had additional bluff combos, our call would begin to make a profit.

If we figured it was a close spot, we might do the combinatorics work. But if we have a read that our opponent never bluffs in a certain spot, we have a clear fold without the need of any combination counting.

Counting Postflop Combinations in Hold’em

Unpaired Holdings

To calculate the remaining combinations of a certain unpaired (preflop) hand, we multiply the number of first card remaining in the deck, with the number of the second card remaining in the deck. To make this clear, let’s see an example.

Example – How many combinations of KJ are there on the following textures?

1) A25
2) K23 
3) KJJ

1) There is no King or Jack out there on the board, meaning there are four Kings and four Jacks left in the deck. 4 * 4 = 16 combinations of KJ.

2) There is a King already out there meaning that there are only three Kings left in the deck. 4 * 3 = 12 combinations of KJ. 

3) There are two Jacks, and three Kings left in the deck. 3 * 2 = 6 combinations of KJ.

Paired Holdings

By “paired” holdings in this context we are referring specifically to pocket pairs. The rules for determining the number of paired combinations left in the deck are a little different. We look at the number of cards remaining of that specific rank and multiply it by “itself minus one” and then divide by two. Let’s see that expressed as a formula.

Where X = the number of cards of a specific rank left in the deck.

Number of combinations of a certain pocket pair  = (x * (x-1)) / 2

Example – How many combinations of 66 are there on the following textures?

1) 552
2) 622
3) 662

1) There are no Sixes on the board texture meaning there are 4 left in the deck. (4 * (4-1))/2 = 6 combinations of 66.

2) There is already one Six out there meaning that there are 3 left in the deck. (3 * (3-1))/2 = 6 combinations of 66.

3) There are already two Sixes out there meaning that there are 2 left in the deck. (2 * (2-1))/2 = 1 combination of 66.

Combinations can be hence used to ascertain the weighting of certain types of hands. For example, it may often surprise players to learn that on boards with a possible straight, there are more combinations of straights than flushes. On an AQT texture there are 16 possible combinations of straights but only 9 combinations of possible sets.

See Also 

Bluff, Bluffcatch, Pot-Odds, Call, Hold’em, Omaha

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