Difference between revisions of "2021 AMC 10A Problems/Problem 20"
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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: | ||
| − | + | \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 <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 16:25, 11 February 2021
Problem
In how many ways can the sequence 
be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?
Solution (bashing)
We write out the 
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 
permutations that work. ~contactbibliophile
