Why two pair beats one pair: counting the combinations

Two pair beats one pair for one reason only, and it is a counting reason rather than a traditional one. Working the arithmetic once is quicker than trusting the chart, and it leaves you with a method rather than a fact.

The one pair count

Pick the rank that pairs: thirteen ways. Pick two of that rank's four suits: six ways. Then pick three other ranks from the remaining twelve, and give each of those three cards any of four suits. Nothing about those last three cards is forced beyond their ranks being distinct and different from the pair.

The two pair count

Now pick two ranks that pair, which is a choice from thirteen taken two at a time rather than thirteen taken one at a time. Each of those ranks contributes two of four suits. Only one card is then free, and it must avoid both paired ranks. Four of the five cards are pinned instead of two.

Pinning more cards always shrinks the count. That is the whole argument, and it is the same argument that separates every adjacent pair of lines on a hand rank chart.

The habit worth keeping

When a hand type feels like it should rank differently, ask how many of its five cards are forced. The hand with more forced cards is rarer and therefore higher. It is a question you can answer in a few seconds without a table in front of you, and it does not depend on remembering which chart you last read.