Counting Poker Hands: The Maths Behind the Rankings

Anyone who has learned the order of poker hands has noticed something curious: the rarer a hand is, the higher it ranks. That is not an accident. The ranking order of five-card hands reflects how often each one turns up, and you can check it with nothing more than careful counting. This article walks through the idea in plain steps, for readers who enjoy mathematics and card games. It is written for adults, and it is not a strategy guide.

The size of the deck

A standard deck has 52 cards, made up of 13 ranks in each of four suits. A poker hand contains five cards, and the order in which you receive them does not matter. Mathematicians call a selection where order does not matter a combination. The number of different five-card hands is written as 52 choose 5, which is calculated like this:

52 × 51 × 50 × 49 × 48 divided by 5 × 4 × 3 × 2 × 1

The top line works out to 311,875,200 and the bottom line to 120, which gives 2,598,960 different hands. Every probability in poker is a count of the hands that fit a description, divided by this total.

Counting a royal flush

A royal flush is the ace, king, queen, jack and ten of a single suit. There are four suits, so there are exactly four royal flushes. The probability of being dealt one is 4 divided by 2,598,960, which is 1 in 649,740. It is one of the rarest hands in the game.

Counting four of a kind

To build four of a kind, choose the rank first. There are 13 ranks, so 13 choices. The four cards of that rank are all used, leaving one more card, which can be any of the other 48 cards in the deck. That gives 13 × 48 = 624 hands. Dividing the total by 624 gives exactly 4,165, so four of a kind appears about once in every 4,165 hands.

Counting a full house

A full house is three cards of one rank plus a pair of another. The steps are:

1. Choose the rank for the three: 13 ways.
2. Choose which three of its four suits: 4 ways.
3. Choose a different rank for the pair: 12 ways.
4. Choose which two of its four suits: 6 ways.

Multiply them: 13 × 4 × 12 × 6 = 3,744. That is six times as many hands as four of a kind, which explains why a full house ranks lower. It appears about once in every 694 hands.

Why rankings follow the counts

This pattern runs all through the list. As the number of possible hands in a category falls, its place in the ranking rises. Poker's order of hands is essentially a table of rarity, which makes it easy to remember once you see the logic. Fewer ways to make a hand means a higher rank.

What these numbers do and do not tell you

Hand frequencies describe how often a hand occurs in a very large number of deals. They do not predict the next deal. Each hand is independent of the last one, and a rare hand is not due just because it has not appeared recently. This is the same idea behind the well-known gambler's fallacy. In real games, other factors such as the number of players, the betting and the cards you cannot see all shape what happens, so the mathematics of a single deal is only part of the picture.

If you would like some background on why games are built with an edge for the operator, House Edge and Odds Explained is a good companion read. For the cultural side of the cards in your hand, A Short History of Playing Cards tells how the deck came to look the way it does.

A reminder

Card games for money are for adults only, and gambling can lead to losses and, for some people, to harm. Set limits, treat any spending as the cost of entertainment, and take a break if play stops being fun. Rules about gambling differ from one place to another.

Frequently asked questions

How many different poker hands are there?

There are 2,598,960 possible five-card hands from a standard 52-card deck.

Why does a royal flush rank so high?

Because only four of those 2,598,960 hands are royal flushes, making it the rarest of all.

Does counting hands help me predict the next deal?

No. Counting describes long-run frequency, not what will happen on any one deal.