Counting One-Pair Hands in a Five-Card Deck
Exactly one pair accounts for 1,098,240 of the 2,598,960 five-card hands. The count excludes every stronger category.
One pair is the most common made category among five-card poker hands, and it is still easy to miscount. A pair somewhere in the five cards accidentally sweeps in two pair, three of a kind, and full houses. Those are different categories. This page counts only the hands whose best description is exactly one pair: two cards of one rank, and three cards of three other ranks, with no second pair among them. The figure describes the deck. It is not a hint about continuing in a Hold'em pot.
The ladder of categories is in Poker Hand Rankings Explained. The royal-flush count, which uses the same total of five-card hands, is in How Rare a Royal Flush Is. Both articles assume a 52-card deck and five cards dealt with order ignored.
The total, once
The number of five-card hands is 52 times 51 times 50 times 49 times 48, divided by 120, which is 2,598,960. Every category lives inside that total. If a subcount does not fit under it, a step was repeated or a stronger hand was smuggled in.
Building exactly one pair
Choose the rank of the pair: 13 options. Choose two suits out of four for that pair: 6 options. That is 78 ways to pick the pair itself.
The other three cards must be three different ranks, chosen from the 12 ranks that are not the pair, or you would have two pair or three of a kind. The ways to choose three ranks from 12 are 12 times 11 times 10, divided by 6, which is 220. Each of those three cards can be any of four suits: 4 times 4 times 4 is 64. They are one card each, so they do not form a pair of their own. They also cannot match the pair's rank, because that rank was excluded.
Multiply. 13 times 6 is 78. 78 times 220 is 17,160. 17,160 times 64 is 1,098,240. So 1,098,240 hands are exactly one pair.
The share, and what sits outside it
1,098,240 divided by 2,598,960 is about 42.3 percent. The honest statement is the two integers. Say 42.3 percent only if you are willing to show them. The rest of the hands are not nothing. They include high card, with no pair, and every stronger category: two pair, three of a kind, straights, flushes, full houses, four of a kind, and straight flushes. A full house contains a pair and is not one pair. Two pair contains pairs and is not one pair. The name is the best name, not a shopping list of ingredients.
What you should not do with 42 percent
Do not hear common as likely to win a pot. Common means many five-card combinations look like this. Several players can hold pairs, and the board can outrank all of them. Do not hear it as the chance your two hole cards are a pocket pair. That is a two-card count, and it is a different number. Do not hear it as a schedule. Each fair deal is a new draw from the combinations, not a queue. Quote the integers or skip the percentage. Do not stake rent on a category because it is crowded. Crowded describes the deck. Your budget describes your month.
Frequently asked questions
Why not let the kickers match? If two kickers matched each other you would have two pair. If a kicker matched the pair you would have three of a kind. The separate ranks and the single suits keep those out.
Does the 42.3 percent include ace-high? No. Ace-high with no pair is a high-card hand. It lives in the rest of the 2,598,960.
Is this the chance I win with one pair in Hold'em? No. Hold'em builds a five-card hand from seven cards and compares people. This count is only a catalog. Winning is a different question, and the catalog does not answer it.