2020 AIME I Problems/Problem 5
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Problem
Solution
Realize that any sequence that works (ascending) can be reversed for descending, so we can just take the amount of sequences that satisfy the ascending condition and multiply by two.
If we choose any of the numbers 
through 
, there are five other spots to put them, so we get 
. However, we overcount some cases. Take the example of 
. We overcount this case because we can remove the 
or the 
. Therefore, any cases with two adjacent numbers swapped is overcounted, so we subtract 
cases (namely, 
,) to get 
, but we have to add back one more for the original case, 
. Therefore, there are 
cases. Multiplying by gives the desired answer, 
.
-molocyxu
See Also
| 2020 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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