Casework is powerful, but bad casework becomes a mess.
The goal is to split the problem so every possibility appears exactly once.
Good Cases Have Two Features
They are complete: every outcome belongs somewhere.
They are non-overlapping: no outcome belongs in two places.
Example Prompt
How many positive two-digit integers have digits that sum to 10?
A natural case split is the tens digit. If the tens digit is 1, the ones digit is 9. If the tens digit is 2, the ones digit is 8. Continue until the tens digit is 9.
COSMO Question
What would be a bad case split for this problem? Why does it make the count harder?