Combinatorics
Overview
Section titled “Overview”Combinatorics problems often reward turning a messy object into a countable structure. Look for symmetry, complementary counting, bijections, recurrence relations, invariants, and extremal choices.
Checklist
Section titled “Checklist”- Define the object being counted before choosing a method.
- Check small cases and search for a stable pattern.
- Look for double-counting identities and bijections.
- Track invariants when a process evolves over time.
- Use extremal arguments when a largest, smallest, first, or last object is forced.
Common moves
Section titled “Common moves”- Count the same set two ways.
- Pair objects with simpler objects.
- Break into cases only after choosing a useful statistic.
- Replace a process with an invariant or monovariant.
Practice problems
Tags: combinatorics
| Done | Problem | Source | Difficulty | Tags |
|---|---|---|---|---|
| AMC 12 2021 #25 | AMC 12 | 2 | combinatorics | |
| Putnam 2023 A1 | Putnam | 3 | combinatorics, number-theory |