Constructive Counting & Overcounting
Build the object step by step, then divide by what you repeated.
Count by describing how to construct each object: choose this, then that. Multiply the choices. If the construction produces each object times, divide by — that division is the whole art.
Classic overcount corrections: circular arrangements divide by (rotations), necklaces by (rotations and flips), multiset permutations by the factorials of the repeats.
When a count feels off by a small integer factor, you have almost certainly built each object more than once.
Train it on Lemma
Constructive Counting & Overcounting 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
- Coloring ArgumentsPaint the board so the impossible shows its color.
- Double CountingCount it twice, set the answers equal.
- All 26 techniques →