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Home » First, calculate the number of affected residents in each neighborhood: - Imagemakers

First, calculate the number of affected residents in each neighborhood: - Imagemakers

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 Use the principle of inclusion-exclusion on adjacency pairs** — but that’s too vast.

📌 Instead, here’s a feasible and rigorous path:

📌 We use **the inclusion of all valid permutations via constructive placement with gap method**, and **recursive enforcement of no adjacent duplicates**.

📌 But again, too complex.

📌 Alternatively, we recognize that this type of problem is best solved via **programming enumeration**, but since it's Olympiad, perhaps the answer is expected via **advanced combinatorial construction**.

📌 Wait — perhaps the intended method is **arranging the letters by generating valid permutations with constraints using factorials and multiplicative corrections**.

📌 But after extensive analysis, the correct and most efficient known method for such problems is **recursive backtracking with memoization**, but we must give a closed-form answer.

📌 Actually, let’s reframe using **the general formula for multiset permutations with no two adjacent identical elements** — but it’s complex.

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