AMC 8 2026 (Problem 17)Four students are seated in a row. They chat with the people sitting next to them, then rearrange themselves so that they are no longer seated next to any of the same people. How many rearrangements are possible?(A) 2\text{(A)}\;2(A)2(B) 4\text{(B)}\;4(B)4(C) 9\text{(C)}\;9(C)9(D) 12\text{(D)}\;12(D)12(E) 24\text{(E)}\;24(E)24Related TopicsCoreToolkit 65 — Casework: Start with the Most Limiting CaseHints (7)Hint 1Let the people from left to right be 111 to 444. So 1,2,3,41,2,3,41,2,3,4.Hint 2Assume the new arrangement is a1,a2,a3,a4a_1,a_2,a_3,a_4a1,a2,a3,a4.In the new arrangement, 111 cannot be next to 222,222 cannot be next to 333, and 333 cannot be next to 444.Base your casework on a2a_2a2.Hint 3Case 1: a2=1a_2=1a2=1a1,1,a3,a4a_1,1,a_3,a_4a1,1,a3,a41↮2 ⟹ a1,1,a3,21\not\leftrightarrow2\implies a_1,1,a_3,21↔2⟹a1,1,a3,22↮3 ⟹ 3,1,a3,2 ⟹ 3,1,4,2✓ ⟹ 1 case2\not\leftrightarrow3\implies3,1,a_3,2\implies3,1,4,2\quad\checkmark\implies1\text{ case}2↔3⟹3,1,a3,2⟹3,1,4,2✓⟹1 caseHint 4Case 2: a2=2a_2=2a2=2a1,2,a3,a4a_1,2,a_3,a_4a1,2,a3,a42↮3,2↮1 ⟹ ×0 cases2\not\leftrightarrow3,\quad2\not\leftrightarrow1\implies\times\quad0\text{ cases}2↔3,2↔1⟹×0 casesHint 5Case 3: a2=3a_2=3a2=3Similar to Case 2 ⟹ 0\implies 0⟹0 cases.Hint 6Case 4: a2=4a_2=4a2=4Similar to Case 1 ⟹ 1\implies 1⟹1 case.Hint 7Ans=1+0+0+1=2\text{Ans}=1+0+0+1=2Ans=1+0+0+1=2Final Answer(A) 222Related Problems (10)AIME I 2025 (Problem 5)AMC 10A/12A 2025 (Problem 8/4)AMC 10A 2025 (Problem 12)AMC 10A 2025 (Problem 16)AMC 10A/12A 2025 (Problem 24/23)AMC 12A 2025 (Problem 18)AMC 10B 2025 (Problem 9)AMC 10B/12B 2025 (Problem 16/13)View all related problems →
