2021 AMC 10A Problems/Problem 20

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Problem

In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing? $\textbf{(A)} ~10\qquad\textbf{(B)} ~18\qquad\textbf{(C)} ~24 \qquad\textbf{(D)} ~32 \qquad\textbf{(E)} ~44$

Solution (bashing)

We write out the $120$ cases. These cases are the ones that work: \begin{align} 1,3,2,5,4 1,4,2,5,3 1,4,3,5,2 1,5,2,4,3 1,5,3,4,2 2,1,4,3,5 2,1,5,3,4 2,3,1,5,4 2,4,1,5,3 2,4,3,5,1 2,5,1,4,3 2,5,3,4,1 3,1,4,2,5 3,1,5,2,4 3,2,4,1,5 3,2,5,1,4 3,4,1,5,2 3,4,2,5,1 3,5,1,4,2 3,5,2,4,1 4,1,3,2,5 4,1,5,2,3 4,2,3,1,5 4,2,5,1,3 4,3,5,1,2 4,5,1,3,2 4,5,2,3,1 5,1,3,2,4 5,1,4,2,3 5,2,3,1,4 5,2,4,1,3 5,3,4,1,2 \end{align} We count these out and get $\boxed{\text{D: }32}$ permutations that work. ~contactbibliophile