--- name: math-thumb-combinatorics description: "Apply Chapter 9 (Combinatorics: Counting Without Exhaustion) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to define the sample space before choosing a counting formula." --- # Combinatorics: Counting Without Exhaustion Use this chapter to help the reader make a checked mathematical decision. All 18 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: What makes outcomes distinct; whether order matters; replacement; capacities; identical versus distinct objects; symmetry action. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **Can the answer be forced without enumeration?** Define boxes and try pigeonhole, or identify unrestricted binary membership. - **Are identical units being distributed?** Use stars and bars only when recipients are labeled and capacities are unlimited. - **Are several cases being combined?** Add disjoint alternatives, multiply staged choices, and subtract intersections when alternatives overlap. - **What makes outcomes distinct?** Decide whether order matters and whether rotation, reflection, or another symmetry identifies descriptions. - **Is a restriction awkward?** Count its complement, or design recurrence states that remember precisely what legal extension depends on. - **Is the object random?** Express the target count as indicators before trying to analyze dependencies. - **Is an identity the goal?** Define one incidence set and count it in two orders. - **Is the exact count enormous?** Use adjacent ratios, logarithms, or a regime-appropriate approximation to expose scale. - **Does the construction split recursively or add component sizes?** Look for Catalan structure or encode the components with a generating function. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Do not divide by a symmetry count unless every orbit has that size. Re-check independence of choices and capacity restrictions before multiplying. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 9.1.3, 9.2.2, 9.2.3 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C09-D01: Choose whether order changes the outcome**: rule 9.2.2. - **C09-D02: Check counts by exhaustive enumeration**: rule 9.1.2. - **C09-D03: Expose unequal symmetry orbits**: rule 9.1.6. - **C09-D04: See the no-match probability lock onto 1/e**: rule 9.3.3. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.