Difference between revisions of "2024 AIME I Problems/Problem 6"
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==Problem== | ==Problem== | ||
| − | + | Consider the paths of length <math>16</math> that follow the lines from the lower left corner to the upper right corner on an <math>8 \times 8</math> grid. Find the number of such paths that change direction exactly four times, as in the examples shown below. | |
==Solution== | ==Solution== | ||
Revision as of 13:56, 2 February 2024
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, as in the examples shown below.
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 (as the minimum is ) and we use Stars and Bars to get 
.
Thus our answer is 
.
