Difference between revisions of "2024 AIME I Problems/Problem 6"
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| + | ==Problem== | ||
| + | An <math>8*8</math> grid is shown. Find the number of paths from the lower-left hand corner to the upper-right hand corner that consist of <math>16</math> grid movements and exactly four “turns.” [REWORD PLZ] | ||
| + | ==Solution== | ||
| + | We divide the path into eight “<math>R</math>” movements and eight “<math>U</math>” movements. Five sections of alternative <math>RURUR</math> or <math>URURU</math> are necessary in order to make four “turns.” We use the first case and multiply by <math>2</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 (as the minimum is <math>1</math>) and we use Stars and Bars to get <math>(7 \choose 5)=21</math>. | ||
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| + | Thus our answer is <math>7*21*2=\boxed{294}</math>. | ||
Revision as of 14:30, 2 February 2024
Problem
An 
grid is shown. Find the number of paths from the lower-left hand corner to the upper-right hand corner that consist of 
grid movements and exactly four “turns.” [REWORD PLZ]
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 
.
