Problem 17 · 2026 AMC 8
Hard
Counting & Probability
casework
Four students sit in a row and chat with the people next to them. They then rearrange themselves so that no one is seated next to anyone they sat next to before. How many such rearrangements are possible?
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Answer: A — 2.
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Hint 1 of 2
Label the original seats 1, 2, 3, 4. The forbidden new-neighbor pairs are exactly the old neighbors: 1-2, 2-3, 3-4. Instead of listing what's banned, flip it — which pairs are allowed to sit together?
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Hint 2 of 2
List the allowed pairs (1-3, 1-4, 2-4) and you'll see they chain together in essentially one way. A valid row is just a path that uses every student through allowed links.
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Approach: build the ‘allowed-neighbor’ chain instead of testing all 24 orders
- The banned new-neighbor pairs are the old ones: 1-2, 2-3, 3-4. Flip to the allowed pairs — everything else: 1-3, 1-4, and 2-4. A legal new row is a path that strings all four students together using only allowed links.
- Trace the allowed links: 2 connects only to 4, and 3 connects only to 1, so 2 and 3 must be the ends. The only way to join them is 2–4–1–3.
- That chain read either direction gives 2–4–1–3 and 3–1–4–2, so there are 2 rearrangements.
- Why this transfers: ‘who may sit/stand next to whom’ problems become much easier as a graph — draw the allowed connections and count paths through all the vertices, rather than testing every permutation against a list of bans.
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