Poker is one of the most popular card games in the world, played by millions of people both online and offline. It’s a game of skill, strategy, and luck, where players try to create the best possible combination of cards in order to win money from their opponents.
But just how many different combinations are there in poker? Let’s explore this question in detail.
Firstly, it’s important to understand that there are different types of poker games. The most common types are Texas Hold’em, Omaha, Seven-Card Stud, and Five-Card Draw. Each game has its own unique rules and strategies, but they all share the same basic principles when it comes to card combinations.
In poker, there are 10 different hands that you can make. These hands are ranked from highest to lowest as follows:
1. Royal Flush – A straight flush from Ace to Ten. 2. Straight Flush – Any five cards of the same suit in sequence. 3. Four-of-a-Kind – Four cards of the same rank. 4. Full House – Three cards of one rank and two cards of another rank. 5. Flush – Any five cards of the same suit.
6. Straight – Any five cards in sequence (but not all the same suit). 7. Three-of-a-kind – Three cards of the same rank. 8. Two Pair – Two cards of one rank and two cards of another rank. 9. One Pair – Two cards of the same rank. 10. High Card – If no player has any pairs or better, then the highest card wins.
So how many different combinations are there? Well, let’s start by looking at how many possible combinations there are for each hand.
For a Royal Flush, there is only one possible combination out of 2,598,960 possible hands.
For a Straight Flush (excluding Royal Flush), there are 10 possible combinations for each suit (since there are 10 possible sequences of five cards in a suit). Therefore, there are 40 possible combinations in total out of 2,598,960 hands.
For Four-of-a-Kind, there are 13 possible ranks to choose from (since each rank can appear four times in a deck). For each rank, there are then 1,277 possible combinations of the remaining card out of the remaining 48 cards. Therefore, there are 624 possible combinations in total out of 2,598,960 hands.
For a Full House, there are again 13 possible ranks to choose from. For each rank chosen, there are then 12 possible ranks for the other pair.
For each pair chosen, there are then 4 possible cards of that rank. Therefore, there are 3,744 possible combinations in total out of 2,598,960 hands.
For a Flush (excluding Straight and Royal Flush), there are 1,277 possible combinations for each suit (since you can choose any five cards from the same suit). Therefore, there are 5,108 possible combinations in total out of 2,598,960 hands.
For a Straight (excluding Straight Flush and Royal Flush), there are again 10 possible sequences to choose from. However, since we can have any suit for each card in the sequence (except for flushes), we need to subtract the number of straight flushes from this number. There are exactly ten straight flushes (one for each suit) that we have already counted earlier.
Therefore we have only counted only one combination per straight flush instead of ten as was required. Thereby making this calculation wrong. The correct answer is given by subtracting the number of straight flushes from the total number of straights which gives us: $10\cdot4^5 -10 =10240-10=10230$ different straights.
For Three-of-a-Kind, there are 13 possible ranks to choose from. For each rank chosen, there are then 4 possible cards of that rank.
There are then 1,098 possible combinations of the remaining cards out of the remaining 48 cards. Therefore, there are 54,912 possible combinations in total out of 2,598,960 hands.
For Two Pair, there are ${13\choose2}$ ways to choose two ranks out of thirteen. For each pair chosen we have four different cards to choose from.
Thereafter we have to choose any card that is not already used in the pairs. There are ${44\choose1}$ ways to do this. Therefore, the total number of Two Pair combinations is ${{13}\choose{2}} \cdot {4\choose2} \cdot {{44}\choose{1}} =123,552$.
For One Pair, there are again 13 possible ranks to choose from. For each rank chosen there are four different cards of that rank.
Thereafter we have to choose three more cards which aren’t already used in the pair or any other combination (therefore a total of $52-4=48$ choices). There are ${48 \choose3}$ ways to do this. Therefore, the total number of One Pair combinations is ${{13}\choose{1}} \cdot {{4}\choose{2}} \cdot {{48}\choose{3}} =1,098,240$.
For High Card hands we need to exclude all other combinations and thus take only $(52-5)$ numbers into account for the first card chosen and $(52-6)$ numbers for the second card chosen etc., down till $(52-9)$ numbers for the ninth card chosen (since five has already been taken). Therefore we get: $\prod_{i=0}^4(47-i)=47\cdot46\cdot45\cdot44\cdot43=1,712,304,160$.
Adding up all the possible combinations for each hand gives us a total of 2,598,960 possible hands in poker.
In conclusion, there are a lot of different combinations in poker. Whether you’re a beginner or an experienced player, it’s important to understand the different hands and their rankings in order to improve your chances of winning.
So next time you sit down at the poker table, remember that there are over 2.5 million possible hands – but only one can be the winner. Good luck!
