2015 AMC 12B Problem 22
Attempt Problem 22 of the 2015 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 12B solutions, or check the answer key.
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22.
Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same chair and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done?
Answer: D
Small Hint:
Relabel so that each person must stay put or move to an adjacent chair
Big Hint:
The number of people who keep their (relabeled) seat must be even: or
Solution:
First imagine everyone moves to the chair directly opposite. The condition becomes: each person must sit in the same chair or an adjacent one. The number of people who keep their seat must be even (otherwise an odd-length gap cannot be filled).
If keep their seat, everyone shifts left, shifts right, or swaps with a neighbor: ways. If keep their seats, those two must be opposite or adjacent, giving choices, and the remaining people are forced to swap in adjacent pairs. If keep their seats, the other two must occupy adjacent seats and swap, giving choices. If all stay, there is way. The total is
Thus, the correct answer is D.
Problem 22 in Other Years
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