Mathematics from zero
Combinatorics: build a counting handbook
Knowing the three rules is not the same as reaching for the right one under pressure. In this project you turn the unit into a reusable tool: a one-page handbook that takes any ‘how many possibilities’ question, routes it to the counting principle, a permutation, or a combination, and proves the count without listing.
Turn the unit’s three rules into a repeatable method: read a counting situation, decide whether choices are independent and whether order matters, apply the matching rule, show the arithmetic, and verify the count by brute-force listing on a small version of the same problem.
Build a one-page 'counting handbook' — a decision flow plus six fully worked real-world counting problems — that demonstrates you can pick and apply the right rule (counting principle, permutation, or combination) and prove each count is exact.
- All six problems are solved with the correct rule and the correct final count, each preceded by the two-question routing sentence.
- Every answer shows its arithmetic step by step, not only the result.
- The two brute-force verifications list every outcome on the small version and the count matches the formula exactly.
- The decision flow at the top, applied to any of the six problems, leads to the rule that problem actually used — no problem contradicts its own routing.
- Add a seventh problem where the naive approach over- or under-counts (such as a round-table seating where rotations are the same, or a password that allows repeated characters) and explain how it bends one of the three rules.
- Add a tiny estimate-vs-exact panel: for one problem, write down a rough guess before computing, then compare it to the exact count and reflect on how fast factorials make intuition fail.
- Turn the handbook into a one-screen flowchart (boxes and arrows) so someone who has not taken the unit could route a new problem to the right rule.
- Pick one problem and compute both its permutation count and its combination count, then write one sentence on what real-world change (adding roles, ranking, or order) would flip the answer from one to the other.
This is the method you will run on every real counting question: route with two questions — separate choices or one set, and does order matter — then apply the matching rule, show the arithmetic, and sanity-check the formula against a tiny brute-force list. Building the handbook once forces the routing to become automatic, so the next time someone asks ‘how many possibilities are there’, you reach for the right rule instead of trying to list them.