Poker is one of the most popular card games in the world, played by millions of people both online and offline. The game involves a combination of skill, strategy, and luck, with players attempting to make the best possible hand by using a combination of their own cards and community cards that are shared with all players.
But just how many possible poker hands are there? As it turns out, the answer is quite complex.
To start with, let’s take a look at how poker hands are ranked. The highest-ranking hand is a royal flush, which consists of the Ace, King, Queen, Jack, and Ten of the same suit. This is followed by a straight flush (five cards in numerical order of the same suit), four-of-a-kind (four cards of the same rank), a full house (three-of-a-kind plus a pair), a flush (five cards of the same suit), a straight (five cards in numerical order), three-of-a-kind (three cards of the same rank), two pairs (two sets of pairs), one pair (two cards of the same rank), and finally high card (when no other hand can be made).
Now that we know how hands are ranked, let’s look at how many possible combinations there are. To do this we need to use some basic combinatorial analysis.
The total number of five-card combinations that can be made from a standard 52-card deck is calculated as follows:
52C5 = 52! / ((52-5)!
* 5!) = 2,598,960
This means that there are just over 2.5 million possible five-card combinations that can be made from a standard deck.
But not all combinations are created equal – some hands are more likely than others. Let’s take a look at each hand individually to see how many possible combinations there are for each:
Royal Flush – There are only four possible royal flushes, one for each suit: spades, hearts, diamonds, and clubs.
Straight Flush – There are 10 possible straight flushes for each suit. For example, in the spades suit, there is the Ace-2-3-4-5 straight flush, followed by the 2-3-4-5-6 straight flush, all the way up to the 10-J-Q-K-A straight flush.
Four-of-a-kind – There are 13 possible ranks for four-of-a-kind (one for each card rank). For each rank, there are then four possible combinations of suits. This means there are a total of 13 x 4 = 52 possible four-of-a-kinds.
Full House – To make a full house, you need to combine three cards of one rank with two cards of another rank. There are 13 possible ranks for the three-of-a-kind component and then 12 possible ranks left over for the pair.
For each combination of ranks, there are then four possible suits for each card. This means there are a total of (13 x 12) x (4 x 4) = 3,744 possible full houses.
Flush – To make a flush you need all five cards to be of the same suit. There are four suits to choose from and once you’ve chosen your suit you then need to choose any five cards from that suit. The total number of combinations is calculated as follows:
4C1 * (13C5 -10C1) = 4 * (1,287 – 10) = 5,108
Straight – To make a straight you need all five cards in numerical order. There are different ways to calculate how many possible straights there are but one method is to start with the highest card and count backwards through all five positions. This gives us:
(10C1)^5 – 4 = 1,277
Three-of-a-kind – To make three-of-a-kind you need three cards of the same rank and two other cards that don’t match. There are 13 possible ranks for the three-of-a-kind component.
Once you’ve chosen your rank, there are then four possible suits to choose from for each of the three cards. For the remaining two cards, there are then 12 possible ranks left over and four suits to choose from for each card. This means there are a total of (13 x 4) x (12C2 x 4C1 x 4C1) = 54,912 possible three-of-a-kinds.
Two Pairs – To make two pairs you need two cards of one rank, two cards of another rank, and a fifth card that doesn’t match either pair. There are 13 possible ranks for the first pair, then 12 possible ranks left over for the second pair.
For each pair combination, there are then four suits to choose from for each card. Finally, once you’ve chosen your pairs and suits, there are then 11 possible ranks left over for the fifth card and four suits to choose from. This means there are a total of (13 x 12) x (4C2 x 4C2) x (11C1 x 4C1) = 123,552 possible two pairs.
One Pair – To make one pair you need two cards of the same rank and three other cards that don’t match. There are again 13 possible ranks for the pair component.
Once you’ve chosen your rank, there are then four suits to choose from for each card in the pair. For the remaining three cards, there are then (12C3) ways to choose their ranks and (4C1)^3 ways to choose their suits. This means there are a total of (13 x 4) x (12C3 x 4C1^3) = 1,098,240 possible one pairs.
High Card – To make a high card hand you simply need five cards that don’t match any other hand. There are (52C5 – 10,200) ways to choose such a hand, which gives us:
2,598,960 – 10,200 = 2,588,760
So there we have it – the total number of possible poker hands is the sum of all the above combinations:
4 + 40 + 52 + 3,744 + 5,108 + 1,277 + 54,912 + 123,552 + 1,098,240 + 2,588,760 =
2 ,598 ,960
This means that there are exactly 2 ,598 ,960 different possible poker hands that can be made from a standard deck of cards.
