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 | + | 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. |
==Solution== | ==Solution== | ||
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For <math>U</math>, we have seven ordered pairs of positive integers <math>(a,b)</math> such that <math>a+b=8</math>. | For <math>U</math>, we have seven ordered pairs of positive integers <math>(a,b)</math> such that <math>a+b=8</math>. | ||
| − | For <math>R</math>, we subtract <math>1</math> from each section ( | + | For <math>R</math>, we subtract <math>1</math> from each section (to make the minimum stars of each section <math>0</math>) and we use Stars and Bars to get <math>{7 \choose 5}=21</math>. |
| − | Thus our answer is <math>7 | + | Thus our answer is <math>7\cdot21\cdot2=\boxed{294}</math>. |
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| + | ==See also== | ||
| + | {{AIME box|year=2024|n=I|num-b=5|num-a=7}} | ||
| + | |||
| + | {{MAA Notice}} | ||
Revision as of 19:17, 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.
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 
.
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
