Every counting-strategy move, in order
The full Chapter 1 pipeline for assembling a hard counting argument from scratch.
Reach for this on a counting problem that resists a single formula -- work through these five moves roughly in order, applying only the ones the problem actually calls for.
The five-step pipeline
Identifying what is being chosen or arranged, from what pool, before touching a formula.
Always first -- identify what is being counted before anything else.
Counting a bigger, easier set where every real outcome is hit the same number of times, then dividing that fixed factor away.
Reach for this when a bigger, easier count hits every real outcome the same number of times.
Counting one set of objects two different, valid ways, proving the two resulting expressions equal.
Use to PROVE an identity by counting one set two ways, not just to compute a count.
Term
bijection
A pairing between two sets that matches every element of one to exactly one element of the other, with nothing left over.
Reach for this when the actual set is hard to count but pairs one-to-one with an easier set.
The finite category every object in a pigeonhole argument is sorted into, rarely handed over explicitly by the problem.
Applies to existence claims ("something must repeat"), not direct counts.
Point table
Graph settings
Calculator