The deck, the hand and the total combinations
A standard poker deck has 52 cards: 13 ranks (2 through 10, J, Q, K, A) in each of 4 suits. A 5-card poker hand, in the definition used by the World Series of Poker and by practically every variant without wild cards, is a selection of 5 distinct cards from the deck in which order does not matter: being dealt A♠ K♠ Q♠ J♠ 10♠ in any sequence is the same hand. That property, order irrelevant and no repetition, is what makes the count a combination, not a permutation, and it is why every formula in this guide uses the notation C(n, k), "n choose k".
The combination formula starts from every way to deal out 5 cards in order (52 × 51 × 50 × 49 × 48, or 52!/47!) and then divides by the 5! = 120 different orders in which the same 5 cards could have come out, since they all count as the same hand. The result is the denominator that shows up in every probability in this guide, the same C(n, k) that shows up, for example, when counting how many lottery number combinations cover a given set of numbers.
C(52,5) = 52! / (5! x 47!) = 2598960- 52!
- every possible order of the 52 cards
- 5!
- divides out the 120 orders the same 5 cards could come out in, they are the same hand
- 47!
- divides out the orders of the 47 cards left out
- 2598960
- the total number of distinct 5-card hands
Why the ranking follows rarity exactly
The table below gathers the exact count of every category (derived field by field in the next section) and the resulting probability. The numbers were calculated independently and match, decimal place by decimal place, the combinatorial probability notes of McGill University’s MATH 323 course, a good sign there is no typo or transposed digit.
| Hand | Combinations | Probability | Odds |
|---|---|---|---|
| Royal flush | 4 | 0.000154% | 1 in 649,740 |
| Straight flush | 36 | 0.001385% | 1 in 72,193.3 |
| Four of a kind | 624 | 0.024010% | 1 in 4,165 |
| Full house | 3,744 | 0.144058% | 1 in 694.2 |
| Flush | 5,108 | 0.196540% | 1 in 508.8 |
| Straight | 10,200 | 0.392465% | 1 in 254.8 |
| Three of a kind | 54,912 | 2.112845% | 1 in 47.3 |
| Two pair | 123,552 | 4.753902% | 1 in 21 |
| One pair | 1,098,240 | 42.256903% | 1 in 2.4 |
| High card | 1,302,540 | 50.117739% | 1 in 2.0 |
| Total | 2,598,960 | 100% | - |
Notice the pattern: reading the table from bottom to top, every category has strictly fewer combinations than the one before it, and that is exactly the order the poker ranking uses. A full house (3,744) beats a flush (5,108) because it has fewer combinations, not because it "feels" stronger; a flush beats a straight (10,200) for the same reason. That is not a coincidence of just any deck: a 2025 mathematics preprint (Williamson, arXiv 2511.06145) studied decks with other numbers of ranks and showed the standard deck, with 13 ranks, is the smallest deck where the raw frequency of every category agrees with the traditional ranking order for all hand types, except within high-card subcases. In other words, the historical choice of 13 ranks per suit is not just cultural, it lands exactly on the point where the math and the game’s convention line up without contradiction.
Deriving each count, the multiplication principle
Every count in this guide uses the same idea: break the hand choice into independent steps (pick ranks, then pick suits) and multiply the number of options at each step. That is the multiplication principle, the same logic as "if there are 3 ice cream flavors and 2 cone types, there are 3 × 2 = 6 combinations". Four of a kind is the simplest case to start with: pick the rank that will repeat 4 times (13 options), the 4 cards of that rank are mandatory (there is only one way to take all 4, C(4,4) = 1), and the fifth card can be any of the remaining 48.
N(quadra) = 13 x C(48,1) = 13 x 48 = 624- 13
- the rank chosen to repeat 4 times
- C(48,1)
- the fifth card, any of the 48 cards left over (52 minus the 4 already used)
- 624
- total four-of-a-kind hands
A full house has one extra step because it has two parts with variable suits: the triple and the pair. First pick the triple’s rank (13 options) and which 3 of its 4 suits go into the hand, C(4,3) = 4 ways. Then pick the pair’s rank among the 12 remaining ranks, and which 2 of its 4 suits go in, C(4,2) = 6 ways.
N(full house) = 13 x C(4,3) x 12 x C(4,2) = 13 x 4 x 12 x 6 = 3744- 13
- rank chosen for the triple
- C(4,3)
- which 3 of the 4 suits of that rank form the triple (4 ways)
- 12
- rank chosen for the pair, any of the 12 remaining ranks
- C(4,2)
- which 2 of the 4 suits of that rank form the pair (6 ways)
The rarest hand of all, the straight flush, uses a different logic: instead of repeating a rank, it is a run of 5 consecutive ranks, all in the same suit. There are 10 possible starting points for a run of 5 ranks in a 13-rank deck: A-2-3-4-5 (the "ace-low" straight), 2-3-4-5-6, 3-4-5-6-7, and so on up to 10-J-Q-K-A, which is exactly the royal flush. So of the 10 starting points, 1 is the royal flush and the other 9 are "plain" straight flushes; multiply each group by the 4 suits.
N(straight flush) = (10 - 1) x 4 = 36 N(royal flush) = 1 x 4 = 4- 10
- starting points of a 5-rank run, from A-2-3-4-5 to 10-J-Q-K-A
- -1
- removes the 10-J-Q-K-A starting point, counted separately as the royal flush
- 4
- possible suits for the run
The remaining categories follow the same recipe, just changing how many cards vary in suit and how many ranks are left free. The table gathers each one’s formula; all were checked by adding up the 10 results and matching the total of 2,598,960.
| Hand | Counting formula | Result |
|---|---|---|
| High card | C(13,5)×4⁵ − flushes − straights + straight flushes | 1,302,540 |
| One pair | 13 × C(4,2) × C(12,3) × 4³ | 1,098,240 |
| Two pair | C(13,2) × C(4,2)² × 44 | 123,552 |
| Three of a kind | 13 × C(4,3) × C(12,2) × 4² | 54,912 |
| Straight | 10 × 4⁵ − 40 (straight flushes) | 10,200 |
| Flush | C(13,5) × 4 − 40 (straight flushes) | 5,108 |
| Full house | 13 × C(4,3) × 12 × C(4,2) | 3,744 |
| Four of a kind | 13 × C(48,1) | 624 |
| Straight flush | 9 × 4 | 36 |
| Royal flush | 1 × 4 | 4 |
Two worked examples: flush and two pair
The two examples below show the full derivation, from scratch to the final probability, for the two most useful hands to understand well: the flush, because it illustrates the trap of double-counting the straight flush, and two pair, because it has one more step than any other category.
- Pick the suit4 possible suits for the whole flush: C(4,1) = 4.
- Pick 5 ranks within that suitThere are 13 cards in that suit; choosing 5 of them is C(13,5) = 1,287.
- Multiply4 × 1,287 = 5,148. That number unintentionally includes every straight flush, which is also 5 cards of the same suit.
- Subtract the straight flushesThere are 40 straight flushes in total (36 plain + 4 royal). 5,148 − 40 = 5,108 "plain" flushes.
- Compute the probability5,108 / 2,598,960 ≈ 0.19654%, or roughly 1 in 509 hands.
- Pick the two ranks that will pair upFrom 13 ranks, pick 2 to become pairs: C(13,2) = 78.
- Pick the suits of each pairFor each of the 2 ranks, pick 2 of the 4 suits: C(4,2) = 6, so 6 × 6 = 36 suit combinations for both pairs together.
- Multiply the two-pair core78 × 36 = 2,808 ways to build the two pairs.
- Pick the fifth cardIt needs a rank different from the two already paired, 11 ranks are left, each with 4 suits: 11 × 4 = 44 options.
- Multiply and compute the probability2,808 × 44 = 123,552. Divided by the total: 123,552 / 2,598,960 ≈ 4.7539%, or roughly 1 in 21 hands.
Wild cards, variants and common ranking mistakes
Everything in this guide holds for the game without wild cards, with a 52-card deck. As soon as a wild card is added (or "wild" cards like deuces enter the rules), the clean correspondence between rarity and ranking stops holding: a wild card can complete any hand, which disproportionately inflates the count of the higher categories (five of a kind, five cards of the same rank, only exists with a wild card) and upends the original rarity order. That is why wild-card tables usually carry an explicit separate rule slotting five of a kind at the top of the ranking, instead of letting the raw count "decide" on its own. Do not confuse this fixed-combination math with blackjack, where there is no "hand ranking" at all, the winning hand is decided by the sum of card values against the house, a sequential-decision problem, not a combination count.
Why suit never breaks a tie in poker
Notice that every formula in this guide multiplies by a suit combination, C(4,k), and those combinations do not favor any particular suit: choosing 2 of the 4 heart suits gives exactly the same 6 options as choosing 2 of the 4 spade suits. Because the deck has exactly 4 symmetric suits, no hand category is more likely in one suit than another, which is why standard poker rules (unlike games such as bridge) never use suit to break a tie between two hands of the same category and the same top rank.
Is three of a kind really weaker than a straight?
Yes, and it is worth looking at the numbers again to believe it: three of a kind has 54,912 combinations (2.11%), almost 5.4 times more than the 10,200 of a straight (0.39%). Many people find this counterintuitive because "3 matching cards" looks more impressive visually than "5 cards in a row", but the count leaves no doubt, a straight is the rarer of the two hands, and the ranking respects exactly that. The rarity here is fixed, hand after hand, unlike the design of other luck-based games, such as the pity system in gacha games, where the chance of a rare item rises with every failed attempt by the game’s own rule.
Frequently asked questions
Why does a full house beat a flush?
What is the rarest hand in poker?
What is the probability of being dealt a pair?
Does the ranking order change in Texas Hold’em?
How can I check these numbers myself?
Does suit break a tie between two equal poker hands?
The poker ranking is not convention, it is rarity in order: of the 2,598,960 possible 5-card hands from a 52-card deck, every category has an exact count derivable by the multiplication principle, from a pair (1,098,240 combinations) to a royal flush (4 combinations), and the smaller the count, the higher the hand ranks. Wild cards and 7-card games change the practical frequency, but they inherit this same order, which remains, point for point, the math of a 52-card deck.