The total number of hands
A poker hand here means five cards from a standard 52-card deck, where the order the cards arrive in does not matter. Holding the ace of spades then the king of hearts is the same hand as holding them in the opposite order.
Start with ordered deals. The first card can be any of 52, the second any of the remaining 51, then 50, 49 and 48:
52 × 51 × 50 × 49 × 48 = 311,875,200.
Any particular set of five cards can be dealt in 5! = 120 different orders, and the ordered count includes every one of them. To count each set once, divide:
311,875,200 ÷ 120 = 2,598,960.
This is written “52 choose 5”, or C(52, 5). If the deck is fairly shuffled and dealt, each of these hands is equally likely, so any one specific hand has a chance of 1 in 2,598,960. The method is the general combinations calculation described in the pillar on counting principles.
How to count a hand type
Every hand type is counted the same way: choose the ranks, choose the suits, and multiply. Ranks are the 13 values (ace through king); each rank has 4 suits.
Four of a kind
Choose the rank of the four matching cards: 13 ways. All four suits of that rank are used, so there is 1 way to pick the suits. The fifth card can be any of the other 48 cards.
13 × 1 × 48 = 624 hands.
The chance is 624 ÷ 2,598,960 = 1/4,165, or about 0.024 percent.
Full house
A full house is three of one rank and a pair of another.
- Choose the rank for the three: 13 ways.
- Choose which 3 of its 4 suits: C(4, 3) = 4 ways.
- Choose a different rank for the pair: 12 ways.
- Choose which 2 of its 4 suits: C(4, 2) = 6 ways.
13 × 4 × 12 × 6 = 3,744 hands. That is about 0.144 percent.
Three of a kind
Exactly three of one rank, plus two cards of two other, different ranks (so it is not a full house).
- Rank of the three: 13 ways, and 4 ways to pick their suits.
- The other two cards have different ranks from each other and from the three: choose 2 ranks from the remaining 12, C(12, 2) = 66 ways.
- Each of those two cards can be any of 4 suits: 4 × 4 = 16.
13 × 4 × 66 × 16 = 54,912 hands, about 2.11 percent.
Flush
A flush is five cards of the same suit. Choose the suit (4 ways) and then 5 of its 13 ranks, C(13, 5) = 1,287 ways.
4 × 1,287 = 5,148 hands. That count includes hands that are also straights in the same suit. There are 40 of those, since a straight can start at ace, 2, and so on up to 10, giving 10 per suit and 10 × 4 = 40. Removing them leaves 5,108 ordinary flushes. Whether those 40 count as flushes depends on the way hands are ranked, so a count should always say how it treats them.
One pair
- Rank of the pair: 13 ways; suits: C(4, 2) = 6 ways.
- Three other cards of three different ranks, none matching the pair: C(12, 3) = 220 ways to pick the ranks.
- Each of those three cards can be any of 4 suits: 4 × 4 × 4 = 64.
13 × 6 × 220 × 64 = 1,098,240 hands, a little over 42 percent of all hands.
Checking the whole picture
Doing this for every category gives a complete partition of the 2,598,960 hands, where each hand falls into exactly one category and the highest category is used:
| Hand type | Number of hands |
|---|---|
| Straight flush (including royal) | 40 |
| Four of a kind | 624 |
| Full house | 3,744 |
| Flush (not a straight flush) | 5,108 |
| Straight (not a straight flush) | 10,200 |
| Three of a kind | 54,912 |
| Two pair | 123,552 |
| One pair | 1,098,240 |
| No pair (high card) | 1,302,540 |
| Total | 2,598,960 |
The numbers add up to 2,598,960. A table that sums correctly is good evidence that no category was double counted or missed, and this is the same check that guards against the problem described by the inclusion-exclusion principle.
For two pair, the same technique gives C(13, 2) × 6 × 6 × 44 = 78 × 36 × 44 = 123,552, where the 44 is the number of cards whose rank differs from both pairs. Straights follow from 10 starting ranks times 4⁵ = 1,024 suit choices, giving 10,240, minus the 40 straight flushes.
What these counts do and do not mean
These are counts of five-card hands dealt from a fresh, fully shuffled deck. They describe the frequency of each type among all possible deals. They do not by themselves describe what happens in a game in which players see other cards, discard and draw, or bet on information; that involves other ideas such as drawing specific cards and the more general poker hand odds. Cards that have already been seen change the counts, and the effect is described by the hypergeometric distribution.
Knowing how rare a hand is does not make a hand “due”, and it does not give a way to profit from card games where there is a built-in disadvantage. Gambling can be harmful, and anyone who feels it is becoming a problem should consider contacting a local support service.