Double Counting
Count it twice, set the answers equal.
Choose a set, count it in two different ways, and equate. The identity that falls out is often one you could never have guessed directly — this is how most combinatorial identities are really proved.
The standard incarnation is counting incidences: sum the degrees of a graph's vertices and you get twice the number of edges. Counting pairs (person, handshake) gives the handshake lemma for free.
When a problem hands you a doubled quantity or an unexpected factor of two, double counting is usually lurking underneath.
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Double Counting unlocks at Level 4 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.
Find your levelMore techniques
- Constructive Counting & OvercountingBuild the object step by step, then divide by what you repeated.
- Modular ThinkingWhen the numbers are too big, shrink the universe.
- All 26 techniques →