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
During AMC testing, the AoPS Wiki is in read-only mode and no edits can be made.

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

(→‎Solution)
(→‎Solution)
 
(One intermediate revision by the same user not shown)
Line 9: Line 9:
 
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.
 
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:
+
Ex. Inserting 4 into the string 123:
−
Putting 4 into the string 123:
+
4 can go before the 2 (1423), before the 3 (1243), and at the very end (1234).
−
4 can go before the 2: 1423,
 
−
Before the 3: 1243,
 
−
And at the very end: 1234.
 
  
 
Only the addition of the next number, <math>n</math>, will change anything.
 
Only the addition of the next number, <math>n</math>, will change anything.

Latest revision as of 11:15, 25 June 2024

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.

Ex. Inserting 4 into the string 123: 4 can go before the 2 (1423), before the 3 (1243), and at the very end (1234).

Only the addition of the next number, $n$, will change anything.

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

Start with $n=3$ since all $6$ permutations work. And go up: $18, 54, 162, 486$.

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

When you are faced with a brain-fazing equation and combinatorics is part of the problem, use recursion! This same idea appeared on another AIME with an 8-box problem.

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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2015_AIME_II_Problems/Problem_10&oldid=221098"