2011 AIME II Problems/Problem 6
Define an ordered quadruple of integers as interesting if , and . How many interesting ordered quadruples are there?
Rearranging the inequality we get . Let , then is a partition of 11 into 5 positive integers or equivalently: is a partition of 10 into 5 non-negative integer parts. Via a standard balls and urns argument, the number of ways to partition 10 into 5 non-negative parts is . The interesting quadruples correspond to partitions where the second number is less than the fourth. By symmetry, there are 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
Let us consider our quadruple (a,b,c,d) as the following image xaxbcxxdxx. The location of the letter a,b,c,d represents its value and x is a place holder. Clearly the quadruple is interesting if there are more place holders between c and d than there are between a and b. 0 holders between a and b means we consider a and b as one unit ab and c as cx yielding ways, 1 holder between a and b means we consider a and b as one unit axb and c as cxx yielding ways, 2 holders between a and b means we consider a and b as one unit axxb and c as cxxx yielding ways and there cannot be 3 holders between a and b so our total is 56+20+4=.
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