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Difference between revisions of "2021 AMC 10A Problems/Problem 20"

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43512,45132,45231,51324,51423,52314,52413,53412. \linebreak</math>
 
43512,45132,45231,51324,51423,52314,52413,53412. \linebreak</math>
 
We count these out and get <math>\boxed{\text{D: }32}</math> permutations that work. ~contactbibliophile
 
We count these out and get <math>\boxed{\text{D: }32}</math> permutations that work. ~contactbibliophile
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== Video Solution by OmegaLearn (Using PIE - Principal of Inclusion Exclusion) ==
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https://youtu.be/Fqak5BArpdc
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~ pi_is_3.14

Revision as of 23:02, 11 February 2021

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: $13254,14253,14352,15243,15342,21435,21534,23154,24153,24351,25143,25341,\linebreak 31425,31524,32415,32451,34152,34251,35142,35241,41325,41523,42315,42513,\linebreak 43512,45132,45231,51324,51423,52314,52413,53412. \linebreak$ We count these out and get $\boxed{\text{D: }32}$ permutations that work. ~contactbibliophile


Video Solution by OmegaLearn (Using PIE - Principal of Inclusion Exclusion)

https://youtu.be/Fqak5BArpdc

~ pi_is_3.14

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