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Home » Now, subtract the number of arrangements where the two N's are adjacent. - inBeat

Now, subtract the number of arrangements where the two N's are adjacent. - inBeat

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Alternate approach: First, arrange the 3 community leaders (including the fixed one) to create gaps.

📌 Fix Leader A at a seat. Then place the other 2 community leaders among the remaining 7 seats, but better: arrange the 3 community leaders so they create slots for scientists.

📌 Better method: Fix Leader A. The remaining 7 seats must be filled with 4 scientists and 3 community leaders, with no two scientists adjacent.

📌 To prevent adjacent scientists, place the 3 community leaders first (total 3 C Dev) around the table with the fixed leader. These 3 create 3 gaps between them (and one opposite the fixed leader), but in linear adjacency (since one seat fixed), we treat the 7 seats as having 3 C Dev placed, creating 4 possible gaps (before, between, after) to place scientists, with at most one scientist per gap.

📌 But we have 3 community leaders and 4 scientists — 7 people. To ensure no two scientists are adjacent, we must place scientists in non-adjacent positions.

📌 Standard method: First arrange the 3 community leaders around the circle (with one fixed), creating 3 gaps between them where scientists can sit. But we have 4 scientists and only 3 gaps — so by pigeonhole, at least one gap has more than one scientist — impossible to avoid adjacency.

📌 But wait: total people: 5 scientists and 3 community leaders. If we fix one person (a community leader), then we have 7 seats left, to place 4 scientists and 3 community leaders.

📌 Therefore, the number of valid seating arrangements is:

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