Module 2: Combinatorics & Graph Theory: Mistake Clinic

This clinic turns wrong moves into reusable judgment. Use it after each practice page and again before the quiz or checkpoint.

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Module-Specific Mistake Radar

Start with these traps. Replace or extend them with real mistakes from your own work.

Mistake to look for	Where it shows up	Symptom	Repair evidence
Finishing Counting and Combinatorial Proof Lab with only a final answer	Counting and Combinatorial Proof Lab	The work has no failed case, trace, test, proof gap, or design stress point.	Add the smallest broken example and show the repair that changes the result.
Finishing Constrained Counting and Recurrence Workshop with only a final answer	Constrained Counting and Recurrence Workshop	The work has no failed case, trace, test, proof gap, or design stress point.	Add the smallest broken example and show the repair that changes the result.
Finishing Graph Structure and Proof Clinic with only a final answer	Graph Structure and Proof Clinic	The work has no failed case, trace, test, proof gap, or design stress point.	Add the smallest broken example and show the repair that changes the result.
Finishing Code Katas with only a final answer	Code Katas	The work has no failed case, trace, test, proof gap, or design stress point.	Add the smallest broken example and show the repair that changes the result.
Treating Counting by Structure, Not by Guessing as vocabulary instead of a tool	Counting by Structure, Not by Guessing	The explanation names the concept but cannot decide between two cases.	Write one example, one non-example, and the rule that separates them.
Treating Permutations, Combinations, and Multinomial Thinking as vocabulary instead of a tool	Permutations, Combinations, and Multinomial Thinking	The explanation names the concept but cannot decide between two cases.	Write one example, one non-example, and the rule that separates them.

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Practice Mistake Checks

Pull any miss from these checks into your mistake log.

Counting and Combinatorial Proof Lab

Source: practice/01-counting-and-combinatorial-proof-lab.md

Explain the error in each move:

1. "At least one forbidden feature" was counted by adding separate cases with no overlap correction.
2. "n! counts all arrangements, so it works even when letters repeat."
3. "C(n,k) and P(n,k) are basically interchangeable for large n."

Constrained Counting and Recurrence Workshop

Source: practice/02-constrained-counting-and-recurrence-workshop.md

Find the flaw in each:

1. Using stars and bars when the distributed objects are distinct.
2. Writing a recurrence from the first four numerical values alone.
3. Applying inclusion-exclusion without defining the underlying sets.

Graph Structure and Proof Clinic

Source: practice/03-graph-structure-and-proof-clinic.md

Explain what is wrong in each claim:

1. "This drawing has crossings, so the graph is nonplanar."
2. "The graph has n - 1 edges, so it is automatically a tree."
3. "All degrees are even, so the graph must have a Hamilton cycle."

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Repair Protocol

For each real mistake:

1. Reproduce the failure on the smallest example, trace, proof, query, command, or design sketch.
2. Name the hidden assumption.
3. Repair the artifact.
4. Save evidence that changed: failing then passing test, corrected proof step, revised diagram, safer command, benchmark, or review note.
5. Add one retrieval card beginning with Check... before... or Do not use... when....

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Mistake Log

Date	Mistake	Symptom	Root cause	Repair evidence	Retrieval card
Starter	Pick one radar row above	Explain how it would fail in this module	Name the assumption	Add a counterexample or corrected artifact	Write the card before closing the page

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Completion Standard

• At least five real mistakes are logged.
• At least two mistakes include a counterexample or failing test.
• At least one mistake connects to an older semester skill.
• At least one correction changes code, a proof, a diagram, a command transcript, a query, or a design decision.