# Difference between revisions of "2011 AIME II Problems/Problem 6"

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Rearranging the inequality we get <math>d-c > b-a</math>. Let <math>e = 11</math>, then <math>(a, b-a, c-b, d-c, e-d)</math> is a partition of 11 into 5 positive integers or equivalently: | Rearranging the inequality we get <math>d-c > b-a</math>. Let <math>e = 11</math>, then <math>(a, b-a, c-b, d-c, e-d)</math> is a partition of 11 into 5 positive integers or equivalently: | ||

− | <math>(a-1, b-a-1, c-b-1, d-c-1, e-d-1)</math> is a partition of 6 into 5 non-negative | + | <math>(a-1, b-a-1, c-b-1, d-c-1, e-d-1)</math> is a partition of 6 into 5 non-negative integer parts. Via a standard balls and urns argument, the number of ways to partition 6 into 5 non-negative parts is <math>\binom{6+4}4 = \binom{10}4 = 210</math>. The interesting quadruples correspond to partitions where the second number is less than the fourth. By symmetry there as many partitions where the fourth is less than the second. So, if <math>N</math> is the number of partitions where the second element is equal to the fourth, our answer is <math>(210-N)/2</math>. |

We find <math>N</math> as a sum of 4 cases: | We find <math>N</math> as a sum of 4 cases: |

## Revision as of 00:08, 2 April 2011

## Problem 6

Define an ordered quadruple as interesting if , and . How many ordered quadruples are there?

## Solution

Rearranging the inequality we get . Let , then is a partition of 11 into 5 positive integers or equivalently: is a partition of 6 into 5 non-negative integer parts. Via a standard balls and urns argument, the number of ways to partition 6 into 5 non-negative parts is . The interesting quadruples correspond to partitions where the second number is less than the fourth. By symmetry there as many partitions where the fourth is less than the second. So, if is the number of partitions where the second element is equal to the fourth, our answer is .

We find as a sum of 4 cases:

- two parts equal to zero, ways,
- two parts equal to one, ways,
- two parts equal to two, ways,
- two parts equal to three, way.

Therefore, and our answer is