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Difference between revisions of "2015 AIME II Problems/Problem 10"

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The simple recurrence can be found.
 
The simple recurrence can be found.
  
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When inserting an integer n into a string with n-1 integers, we notice that the integer n has 3 spots where it can go: before n-1, before n-2, and at the very end.
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When inserting an integer <math>n</math> into a string with <math>n - 1</math> integers, we notice that the integer <math>n</math> has 3 spots where it can go: before <math>n - 1</math>, before <math>n - 2</math>, and at the very end.
  
 
EXAMPLE:  
 
EXAMPLE:  

Revision as of 20:43, 17 December 2015

Problem

Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the integers $1, 2, \ldots, 7$.

Solution

The simple recurrence can be found.

When inserting an integer $n$ into a string with $n - 1$ integers, we notice that the integer $n$ has 3 spots where it can go: before $n - 1$, before $n - 2$, and at the very end.

EXAMPLE: Putting 4 into the string 123: 4 can go before the 2: 1423, Before the 3: 1243, And at the very end: 1234.

Thus the number of permutations with n elements is three times the number of permutations with $n-1$ elements.

However, for $n=1$, there's an exception: there's only 2 places the 2 can go (before or after the 1).

For $n=1$, there are $2$ permutations. Thus for $n=7$ there are $2*3^5=\boxed{486}$ permutations.

See also

2015 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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