Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

(→‎Solution (bashing))
(→‎Solution (bashing))
Line 10: Line 10:
 
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
 +
 +
 +
== Video Solution by OmegaLearn (Using PIE - Principal of Inclusion Exclusion) ==
 +
https://youtu.be/Fqak5BArpdc
 +
 +
~ pi_is_3.14

Revision as of 22:02, 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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2021_AMC_10A_Problems/Problem_20&oldid=145634"