Understanding Poker Hand Combinations – A Comprehensive Overview

Have you ever felt a rush of anxiety when faced with a substantial bet on the river, holding just a bluff catcher? It's common to sense that there's a structured approach we ought to follow, yet our thoughts may get tangled up. blank .

What is it that makes successful players appear to consistently make sound decisions in high-pressure situations? Are we perhaps overlooking a crucial element?

This is where an understanding of poker combinations becomes invaluable. Grasping the intricacies of hand combinations allows us to better assess the likelihood that our current hand holds value. In this piece, we'll explore the following aspects -

Table of Contents

What Are Combinations?

It's important to note there are two closely related, yet distinct, concepts sharing the same name.

The definitions are as follows:

Combinations– Meaning

1. A specific instance of a poker hand (which typically refers to the two hole cards).
2. The mathematical aspect. This considers a selection of items from a group where the order is not significant.

Hand Combinations in Poker

The word 'combination' (or the abbreviated form 'combo') refers to the various ways in which a particular poker hand can be constructed.

For instance: If we get dealt AK during the preflop phase of Hold’em, how many unique AK combinations do we have?

If we wanted, we could enumerate every possible method to form AK. However, seasoned players typically recall that there are 16 combinations for every unpaired hand –

12 are off-suit, and 4 are suited.

Consider this example: Being dealt 66 preflop in Hold’em, what are the different combinations of 66?

Once more, we could go through all possible variations for 66. But it's much simpler to keep in mind that there are 6 combinations for each pocket pair in the game. Holdem .

Calculating Combos in Hold’em

It’s easy enough to remember preflop However, things can become a bit trickier when the card removal effect enters the picture, as certain cards are already known, thereby reducing the potential combinations of various starting hands.

Having a systematic approach for calculating poker combos can be quite beneficial.

Let’s start with unpaired hands.

Combos of Unpaired Hands:

Card 1: Available Cards Count * Card 2: Available Cards Count

Referring back to our situation with AK preflop, we can clearly see how we calculated the total of 16 combinations.
With a standard deck, there are 4 Aces and 4 Kings available for play.

Therefore:  4 * 4 = 16 total combos of AK

Let's tackle a slightly more challenging scenario where we must factor in existing cards. card removal effects .

Example: In Hold’em, if the flop shows Ac8d7h, how many combinations of AK do we have?

The critical aspect of this question lies in recognizing that only 3 Aces remain in the deck, as one has already appeared on the flop.

As such, we have 3 available Aces and 4 available Kings .

Therefore: 3 * 4 = 12 total combos of AK.

Now, let’s delve into the rule regarding pocket pairs.

Combos of Unpaired Hands:
Available Cards Count * (Available Cards Count - 1)

------------------------------------------------------
2

Returning to our preflop example with pocket Sixes, there are four available Sixes in the deck.

Thus: (4 * 3) / 2 = 6 combinations of pocket Sixes.

Let's again consider a slightly tougher question where we need to consider the card removal effect.

Practice Questions with Combos

Now, we should be capable of addressing more complex inquiries regarding combinations, relying on basic math skills and applying common sense.

Example: If we possess pocket Aces preflop, how many combinations of QQ+/AK might our opponent have?

Let’s set aside the card removal effect for now. We can reasonably assume our opponent might possess the following:

QQ+ = 18 combos (6 * 3)
AK = 16 combos
- for a total of 34 combos.

However, because our opponent holds two of the Aces, this will notably affect the number of combinations due to card removal.

This phenomenon is often referred to as the “blocker effect” . The combinations for QQ and KK will still total 6 each; however, AA and AK combos will be affected.

Since we hold two Aces, only one Ace combination is left in the deck.

(2 * 1) / 2 = 1 combo of Aces

There are also 8 combinations of AK (given 2 Aces and 4 Kings are left).

4 * 2 = 8 combos of AK

Thus, compiling all the combinations yields the following:

AA – 1 combo
KK – 6 combos
QQ – 6 Combos
AK – 8 Combos
for a total of 21 combos.

What are the different ways to achieve a set?

It's helpful to remember that there are consistently 3 combinations for each type of set accessible. In total, this amounts to 9 combinations of sets.

Now let's present a slightly more intricate challenge.

Example: In Hold’em, with the flop as Ac8d7h, how many different configurations exist for top pair?

In this context, logic and common sense become vital. We already understand that any top pair hand like AJ yields 12 combos (4 * 3 = 12).

However, we need to identify how many Ax hands can produce a top pair.

Given that there are 13 card ranks in a standard deck, and if we exclude AA, A8, and A9, we have 10 distinct Ax hands left that can form a top pair, each consisting of 12 combinations.

This results in a grand total of 120 (10 * 12) different top combinations. pair on this texture.

Combinations in Mathematics

Now let’s shift our focus to the second interpretation of combinations.

Mathematics: A selection of cards from a hand where the sequence is irrelevant.

Up to this point, we have examined methods for computing combinations. hole card This second interpretation will assist us in determining the probabilities associated with various board outcomes.

Let's modify our definition to fit the realm of poker -

In poker: A selection of cards from a deck where order is not a factor.

Let’s start with an example question.

It is commonly cited that there are approximately 19,600 potential flops in Hold’em.

This can be validated mathematically using combinations.

Let's initiate this by reviewing the mathematical formula for combinations.

Now, let’s clarify the components of this formula -

n = the total number of options available (i.e., the total number of cards in the deck)
r = the total number of selections you wish to make (i.e., the number of cards drawn from the deck)
C  = stands for combinations
! = a mathematical function referred to as 'factorial'. (For instance, 5! equals 5 * 4 * 3 * 2 * 1)

Let’s assign values to n and r -

n = 52 as there are 52 cards in a full deck.
r = 3 since we are drawing 3 cards to form a flop.

Now let's input our values into the formula –

52! 
-----
(49!) * 3!

These figures can be easily inserted into a formula, calculator (with the correct parentheses). However, you can also opt for a manual approach to simplify the formula.

While the method for simplification isn't the main focus of this article, it's not particularly challenging either. We can transform the original formula into this:

52 * 51 * 50
-----------------
6

The outcome suggests there are 22,100 potential flop combinations, which is quite fascinating. So, why does a quick internet search indicate there are only 19,600 possible flop combinations?

Could it be that those figures have been adjusted to account for the two known cards we possess (i.e., our hole cards)?

Let’s rerun the calculation -

n = 50 because there are 50 cards still in the deck after our hole cards have been dealt.
r = 3 because we are drawing 3 cards for the flop.

This time we get –

50!
-----
47! * 3!

Which simplifies to:

50 * 49 * 48
------------------
6
- which equals 19,600!

When players mention there are 19,600 possible flops, they are referring to the situation where 2 cards from the deck are already revealed. We have determined that the total number of flops observable to an outsider is 22,100, as long as they are unaware of the hole cards held by the players.

Now, let’s leverage our combinatorial knowledge to tackle a slightly more complex inquiry.

Example: If we receive ThJh in Hold’em, what are the odds of flopping a flush?

We already know the count of distinct flops (19,600). Our next objective is to find out how many combinations can yield a flush of hearts. flush This calculation presumes that there are still 11 hearts remaining in the deck.

How do we determine the various combinations of 3 cards from a total of 11?

Once again, this type of calculation is exactly what the concept of combinations is intended to handle.

n = 11 because this is the count of hearts available in the deck assuming we possess 2 hearts in hand.
r = 3 since these are the number of cards drawn to create the flop in Hold’em.

Let’s plug these numbers into our formula –

11!
----------
8! * 3!

- which simplifies to

11 * 10* 9
---------------
6
= 165 ways of 3-heart flops being dealt.

Thus, the probability of drawing 3 hearts is –

165 / 19,600 = 0.0084 or 0.84%

When we consult equity calculation software, it informs us that the probability of flopping a flush is 0.82% instead of 0.84%.

Can you see what the discrepancy is?

Some of those heart flops will grant us a straight flush . Understanding how many of those there are is essential so we can subtract them from our total flush flops.

Flops that lead to a straight flush featuring ThJh –

AhKhQh
KhQh9h
Qh8h9h
7h8h9h

That translates to four specific flops we need to subtract from our current count of 165.

161 / 19,600 = 0.0082 or 0.82%

We have now confirmed 0.82%, which aligns with the figure given by our equity calculation software.

Let’s try one more.

Example: If we hold AKo, what is our possibility of flopping a straight?

It seems we are beginning to recognize a pattern with the streamlined version of the formula.

The pattern is as follows –

Number of Card 1 * Number of Card 2 * Number of Card 3
-------------------------------------------------------------------------
3!

When it comes to achieving a straight, we need to hit a flop of TJQ (the order of the cards is irrelevant).

• There are 12 combinations for hitting a T, J, or Q on the flop.
• Assuming we hit a T, we have 8 ways to then draw either a J or Q on the flop.
• Regardless of which flop card we get, there are still 8 ways to catch part of the straight on the turn.

There are 4 options for the river card that would complete our straight.

12 * 8 * 4
---------------
3!

Multiplying out the top portion of the formula (12 * 8 * 4) yields 384. This figure reflects the total distinct ways to make a straight if we consider the order of the cards (which we shouldn’t in this case).

So 384 implies that TcJhQd and JhTcQd are different flops—though they actually aren't for this particular calculation.

The mathematical term for a grouping where order matters is permutation rather than combination . It employs a slightly different formula.

The second segment of the formula (divide by 3! or 6) is meant to prevent counting duplicate flops (where only the order changes).

384
------
6
= 64

This results in 64 distinct ways of forming a straight without regard to the order of flop cards.

Therefore: 64 / 19600 = 0.003265 or 0.33%

While employing combinations to solve this problem isn't strictly required, we can validate the outcome using basic probability calculations.

Let’s see how it works –

1. Event 1 – The first flop card is either a T, J, or Q = 12/50 since 50 cards remain in the deck.
2. Event 2 - The second flop card is one of the remaining 8 cards completing the straight = 8/49 because 49 cards are left in the deck.
3. Event 3 – The third flop card must be one of the 4 cards remaining to finalize the straight = 4/48 since 48 cards are in the deck.

We now multiply the probabilities of each event to produce the overall probability.

12/50 * 8/49 * 4/48 = 0.003265

Look familiar?

This matches the previously calculated value for flopping a straight when we hold AK preflop .

Should we call or fold?

According to our pot odds, we need to win at least 33% of the time to justify a call against a pot-sized wager.

Do we have a reasonable expectation of winning that frequently? Fortunately, we aren't completely in the dark here.

Do we Have the Winning Combination?

Even with a solid grasp of the hand combinations our rival has, it's almost impossible to be entirely certain if we possess the winning hand. We can only achieve absolute certainty if we have the unbeatable hand (and even then, a tie can occur).

So, when we can’t be completely sure whether our hand is the winner, how do poker combinations assist us in making tough decisions? bluffcatch decisions?

Here’s a brief instance of the type of analytic work that can be conducted using combinatorial strategies.

We are in a tricky situation on the river facing a pot-sized bet, and we guess our opponent has around 40 combinations available, 15 of which are bluffs and the remaining 25 are strong hands. Should we call or fold?

Based on our pot odds, we need to win at least 33% of the time to justify a call when confronted with a pot-sized bet. Are our expectations of winning that often justified? Thankfully, we have clarity in this situation. Based on our evaluations of the number of winning combinations from the opponent, we believe we'll have the best hand 15 out of 40 times, or 37.5%.

While that leads us to think we'll end up losing the majority of the time when we call, mathematically speaking, it’s a decision that should yield profits over time, making the call worthwhile. The more adept we are at understanding our opponent's potential combinations, the more accurate and profitable our strategic choices will be.

Hand Combinations Summary

The most frequently referenced type of poker combinations is the hand combinations previously discussed. Typically, this is what players refer to when discussing combinations.

In contrast, very few players are familiar with mathematical combinations . The average player may not have even learned about them, much less how to compute them.

We've discovered that there are 19,600 distinct flops when accounting for two known hole cards. We can leverage poker combinations to determine how often specific flop types appear. The result can then be divided by 19,600 to identify the corresponding probabilities .

A solid understanding of these mathematical concepts can provide a significant advantage in the game.