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.
Solution
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
See also
| 2024 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
