1998 AIME Problems/Problem 7
Let be the number of ordered quadruples of positive odd integers that satisfy Find
Define . Then , so .
So we want to find four integers that sum up to 51; we can imagine this as trying to split up 51 on the number line into 4 ranges. This is equivalent to trying to place 3 markers on the numbers 1 through 50; thus the answer is , and .
Another way we can approach this problem is by imagining a line of 98 stones. We want to place the stones into boxes so that each box has an odd number of stones. We then proceed by placing one stone in each box to begin with, ensuring that we have a positive number in every box. Now we have stones left. Because we want an odd number in each box, we pair the stones, creating sets of . Every time we add a pair to one of the boxes, the number of stones in the box remains odd, because (an odd #) + (an even #) = (an odd #).
Our problem can now be restated: how many ways are there to partition a line of stones? We can easily solve this by using sticks to separate the stones into groups, and this is the same as arranging a line of sticks and stones. Our answer is therefore
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