How Many Possible Hands Are There in a Five-Card Poker Game

Introduction

If you are a new Poker player, you need to understand the fundamentals of the card game before joining the tables. One of the most prominent aspects of the game is card combinations, also called ‘hand’.

This article will explore the mathematics behind the number of possible hands in a standard five-card poker game.

The Objective of a Poker Hand

A set of five cards is dealt to each player from a standard deck. To win the game, one has to aim for the best possible hand according to the hand ranking, including pairs, flushes, straights, and other combinations.

The Mathematics Behind Poker Hands

A standard deck of cards comprises 52 cards in four suits: clubs, diamonds, hearts, and spades. 

Each suit possesses a set of 13 cards, with an Ace being the highest-ranking card and 2 being the lowest.

Total Number of Possible Hands in Poker

We use combinatorial mathematics, specifically ‘combinations’, to determine the number of possible five-card hands. A combination is a way of selecting items from a larger pool where the order does not matter.

The formula for combinations is given by:

C(n,k)=n!/k!(n−k)!

where:

• n is the total number of items.
• k is the number of items to choose from.
• ! denotes factorial, the product of all positive integers up to that number.

Applying the Formula

In the context of a five-card poker hand:

• n=52 (the total number of cards).
• k=5 (the number of cards in a hand).

Plugging these values into the formula gives:

C(52,5) = 52!/5!(52−5)​

C(52,5) = 52!/5!(47)!​

C(52,5) = 52×51×50×49×48/5x4x3x2x1

Performing the multiplication and division, we get:

C(52,5) = 311,875,200/120 = 2,598,960

There exist 2,598,960 unique arrangements of five cards that can be distributed from a deck of 52 cards.

The Impact of Card Order

It's important to note that in poker, the order of cards in a hand does not matter while calculating the number of possible hands. For example, the hand 𝐴♠ 𝐾♠ 𝑄♠ 𝐽♠ 10♠  (a royal flush) is considered identical to 10♠ 𝐽♠ 𝑄♠ 𝐾♠ 𝐴♠ . Therefore, while there are many ways to arrange any set of five cards, these arrangements do not create different hands from a poker perspective.

Number of Possible Hand in Different Variants of Poker 

1. Texas Hold'em

In Texas Hold'em, each player is dealt two private cards, and five community cards are dealt face-up on the board. Players use the best five-card combination of their two private cards and the five community cards.

Total Possible Hands:

• Private hands (2 cards out of 52): 

52!/2!(52-2)! = 52x51/2x1 = 1326

• Five-card combinations from seven cards (52 choose 7 minus irrelevant hands): ≈133,784,560

2. Omaha

In Omaha, each player is dealt four private cards, and five community cards are dealt face-up. Players must use exactly two of their private cards and three of the community cards.

Total Possible Hands:

• Private hands (4 cards out of 52) =270,725
• Five-card combinations from nine cards (considering the use of exactly two hole cards and three community cards): ≈ 60,095,035,636

3. Seven Card Stud

Each player is dealt seven cards throughout the hand (three down, four up) in Seven Card Stud. The best five-card handout of these seven is used.

Total Possible Hands:

• Seven cards from 52=133,784,560
• Five-card combinations from these seven cards: 21
• Therefore, the total combinations: 133,784,560×21 = 2,809,476,760

4. Five Card Draw

In Five Card Draw, each player is dealt five private cards with no community cards. Players can exchange up to three of their cards for new ones from the deck.

Total Possible Hands:

• Five cards out of 52 = 2,598,960

5. Razz (Seven Card Stud Low)

Razz is a lowball form of Seven Card Stud, where the lowest hand wins. Each player is dealt seven cards, and the best five-card low hand is used.

Total Possible Hands:

• Seven cards from 52 = 133,784,560
• Five-card low combinations from these seven cards = 21
• Therefore, the total combinations: 133,784,560×21 = 2,809,476,760

6. Short Deck Hold'em (Six Plus Hold'em)

In Short Deck Hold'em, the deck is reduced to 36 cards (removing cards 2 through 5). Players are dealt two private cards, and five community cards are dealt face-up.

Total Possible Hands:

• Private hands (2 cards out of 36) = 630
• Five-card combinations from seven cards (36 choose 7 minus irrelevant hands): ≈ 29,793,840

Summary of Hand Combinations

These numbers reflect the theoretical total possible hands in each variant, accounting for the various ways hands can be dealt and the rules of each game.

Importance in Poker Strategy

To determine the likelihood of drawing particular hands during gameplay, players must have a basic understanding of the overall number of poker hands. This comprehension enables them to make informed decisions about betting, calling, raising, or folding based on how strong their hand is in comparison to potential opponents.

Frequently Asked Questions

What is the total number of possible hands in Poker?

There are exactly 2,598,960 possible unique combinations of five cards that can be dealt from a standard 52-card deck.

Are all possible hands equally likely to occur?

The possibility of different hands varies, with some being more common than others. For example, the probability of getting a One Pair is much higher than a rare hand like Royal Flush. Hence not all possible hands have an even chance of occurring.

What is the rarest hand in Poker?

With a rarity of about 0.000154%, the Royal Flush is considered to be at the pinnacle of conventional poker games. It comprises A, K, Q, and J, as well as 10, all belonging to one particular suit.

Can the number of possible hands change based on the number of players?

The total number of possible hands (2,598,960) does not change regardless of the number of players involved in the game. Nonetheless, the number of cards each player gets and the number of participants who enter the game will affect the allocation of hands among them.

Conclusion

The attraction of poker goes beyond its tactics. It is rooted in the extensive selection of hands that can be distributed on the table. Since each hand category possesses a unique possibility of emerging, it strongly influences the results and diverse approaches employed during gameplay.