Did We Find Them All?
A difficult universe can be partitioned into cases that are exhaustive, non-overlapping, and easier to count.
What you'll learn
- Recognize why sorting a partial list does not prove completeness.
- Choose a useful feature to split possibilities into smaller groups.
- Finish every group, check that nothing is missing or repeated, and combine the counts.
Before you start
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Worked recap: Make cases that cover every outcome once
Count two-digit numbers using only 1 and 2. Separate the cases by the tens digit: 11 and 12 begin with 1; 21 and 22 begin with 2.
Every allowed number belongs to exactly one case. The two cases contribute 2 + 2 = 4 numbers, with no overlap and nothing missing.