AMC10 2021 10A, Problem 20

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?

Solution: We write out the $5!=120$ cases, then filter out the valid ones: $13254,14253,14352,15243,15342,21435,21534,23154,24153,24351,25143,25341,\linebreak  31425,31524,32415,32514,34152,34251,35142,35241,41325,41523,42315,42513,\linebreak 43512,45132,45231,51324,51423,52314,52413,53412.$ We count these out and get $\fbox{32}$ permutations that work.

Answer: 32