2024 AIME I Problems/Problem 6

Problem

Consider the paths of length $16$ that follow the lines from the lower left corner to the upper right corner on an $8 \times 8$ grid. Find the number of such paths that change direction exactly four times.

Solution

We divide the path into eight “ $R$ ” movements and eight “ $U$ ” movements. Five sections of alternative $RURUR$ or $URURU$ are necessary in order to make four “turns.” We use the first case and multiply by $2$ .

For $U$ , we have seven ordered pairs of positive integers $(a,b)$ such that $a+b=8$ .

For $R$ , we subtract $1$ from each section (to make the minimum stars of each section $0$ ) and we use Stars and Bars to get ${7 \choose 5}=21$ .

Thus our answer is $7\cdot21\cdot2=\boxed{294}$ .

2024 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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2024 AIME I Problems/Problem 6

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