2024 AIME I Problems/Problem 6

Problem

Consider the paths of length that follow the lines from the lower left corner to the upper right corner on an grid. Find the number of such paths that change direction exactly four times.

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Solution 1

We divide the path into eight “” movements and eight “” movements. Five sections of alternative or are necessary in order to make four “turns.” We use the first case and multiply by .

For , we have seven ordered pairs of positive integers such that .

For , we subtract from each section (to make the minimum stars of each section ) and we use Stars and Bars to get .

Thus our answer is .

~eevee9406

Solution 2

The path can either start by going up or start by going right. Suppose it starts by going up. After a while, it will turn to the right. Then, it will go up. After that, it will go right again. There are ways to choose when it will go up in the middle of the path and there are to choose the two places it will go right. Thus, there are ways to create a path that starts by going up. By symmetry, this is the same as the number of paths that start by going right, so the answer is

~alexanderruan

Solution 3

Notice that the case and the case is symmetrical. WLOG, let's consider the RURUR case.

Now notice that there is a one-to-one correspondence between this problem and the number of ways to distribute 8 balls into 3 boxes and also 8 other balls into 2 other boxes, such that each box has a nonzero amount of balls.

There are ways for the first part, and ways for the second part, by stars and bars.

The answer is .

~northstar47

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See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.