Technique Encyclopedia

Casework

What it is

Casework means dividing a problem's possibilities into distinct scenarios that together cover every possibility exactly once, then solving (usually counting) each scenario on its own and adding the results.

Signals that suggest using it

A direct count seems to require tracking several independent conditions at once.
The problem has a small, identifiable number of qualitatively different scenarios (e.g. by parity, by which value is largest, by a yes/no condition).
"How many ways..." questions where a single formula doesn't obviously apply.

When it's effective

Casework works best when the cases are few, cleanly defined, and non-overlapping — each one becomes its own small, manageable counting problem. It's often the most reliable method when no elegant shortcut is visible.

When it's not effective

If the natural cases are numerous, overlapping, or hard to define cleanly, casework becomes slow and error-prone. In that situation, complementary counting or a bijection to a simpler set is usually faster and safer.

Simple example

Problem

How many two-digit numbers have a units digit that is either 0 or 5?

Solution

Split into two cases. Units digit 0: the tens digit can be 1–9, giving 9 numbers (10, 20, ..., 90). Units digit 5: the tens digit can again be 1–9, giving 9 more (15, 25, ..., 95). The cases don't overlap, so the total is 18.

AMC-style example

Problem

Three-digit numbers abc (a three-digit number) (with a ≠ 0) are chosen so that a+b+c=6. How many such numbers have a=1?

Solution

With a=1 fixed, we need b+c=5 with b,c digits from 0–9. Casing on b from 0 to 5 (since c = 5 − b ≥ 0 requires b ≤ 5), each value of b gives exactly one valid c, for 6 numbers total: 105, 114, 123, 132, 141, 150.

Complementary Counting Parity Pigeonhole Principle