2015 AIME II Problems/Problem 10
Problem
Call a permutation of the integers quasi-increasing if for each . For example, 53421 and 14253 are quasi-increasing permutations of the integers , but 45123 is not. Find the number of quasi-increasing permutations of the integers .
Solution
The simple recurrence can be found.
When inserting an integer into a string with integers, we notice that the integer has 3 spots where it can go: before , before , 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.
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 elements.
Start with since all permutations work. And go up: .
Thus for there are 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 a 8-box problem.
See also
2015 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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