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, as in the examples shown below.

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 (as the minimum is $1$ ) and we use Stars and Bars to get $(7 \choose 5)=21$ .

Thus our answer is $7*21*2=\boxed{294}$ .

Page

Toolbox

Search

2024 AIME I Problems/Problem 6

Problem

Solution