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?

Let the people from left to right be to . So .
Assume the new arrangement is .
In the new arrangement, cannot be next to ,
cannot be next to , and cannot be next to .
Base your casework on .
In the new arrangement, cannot be next to ,
cannot be next to , and cannot be next to .
Base your casework on .
Case 1:
Case 2:
Case 3:
Similar to Case 2 cases.
Similar to Case 2 cases.
Case 4:
Similar to Case 1 case.
Similar to Case 1 case.
(A)
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