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

Line 6: Line 6:
 
We write out the <math>120</math> cases.  
 
We write out the <math>120</math> cases.  
 
These cases are the ones that work:
 
These cases are the ones that work:
<math>13254,
+
<math>13254,14253,14352,15243,15342,21435,21534,23154,24153,24351,25143,25341,31425,31524,32415,32451,34152,34251,35142,35241,41325,41523,42315,42513,43512,45132,45231,51324,51423,52314,52413,53412,</math>
14253,
 
14352,
 
15243,
 
15342,
 
21435,
 
21534,
 
23154,
 
24153,
 
24351,
 
25143,
 
25341,
 
31425,
 
31524,
 
32415,
 
32451,
 
34152,
 
34251,
 
35142,
 
35241,
 
41325,
 
41523,
 
42315,
 
42513,
 
43512,
 
45132,
 
45231,
 
51324,
 
51423,
 
52314,
 
52413,
 
53412,</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

Revision as of 16:32, 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: $13254,14253,14352,15243,15342,21435,21534,23154,24153,24351,25143,25341,31425,31524,32415,32451,34152,34251,35142,35241,41325,41523,42315,42513,43512,45132,45231,51324,51423,52314,52413,53412,$ We count these out and get $\boxed{\text{D: }32}$ permutations that work. ~contactbibliophile