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 , and 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. 125 lessons, 1206 curated problems and unlimited generated practice at six difficulties. Free to start, no card, and every paid plan opens with 3 free days.
Find your levelMore techniques
- Coloring ArgumentsPaint the board so the impossible shows its color.
- Double CountingCount it twice, set the answers equal.
- All 29 techniques →