2024 AIME I Problems/Problem 6
Contents
[hide]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 1
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
Solution 2
Draw a few examples of the path. However, notice one thing in common - if the path starts going up, there will be 3 "segments" where the path goes up, and two horizontal "segments." Similarly, if the path starts going horizontally, we will have three horizontal segments and two vertical segments. Those two cases are symmetric, so we only need to consider one. If our path starts going up, by stars and bars, we can have ways to split the 8 up's into 3 lengths, and have to split the 8-horizontals into 2 lengths. We multiply them together, and multiply by 2 for symmetry, giving us
~nathan27 (original by alexanderruan)
Solution 3
Notice that the case and the case is symmetrical. WLOG, let's consider the RURUR case.
Now notice that there is a one-to-one correspondence between this problem and the number of ways to distribute 8 balls into 3 boxes and also 8 other balls into 2 other boxes, such that each box has a nonzero amount of balls.
There are ways for the first part, and ways for the second part, by stars and bars.
The answer is .
~northstar47
Feel free to edit this solution
Solution 4
Starting at the origin, you can either first go up or to the right. If you go up first, you will end on the side opposite to it (the right side) and if you go right first, you will end up on the top. It can then be observed that if you choose the turning points in the middle grid, that will automatically determine your start and ending points. For example, in the diagram if you choose the point and , you must first move three up or two right, determining your first point, and move 5 up or 3 right, determining your final point. Knowing this is helpful because if we first move anywhere horizontally, we have points on each column to choose from and starting from left to right, we have points on that row to choose from. This gives us which simplifies to . The vertical case is symmetrical so we have
~KEVIN_LIU
Solution 5
As in Solution 1, there are two cases: or . We will work with the first case and multiply by at the end. We use stars and bars; we can treat the s as the stars and the s as the bars. However, we must also use stars and bars on the s to see how many different patterns of bars we can create for the reds. We must have bar in blacks, so we use stars and bars on the equation . However, each divider must have at least one black in it, so we do the change of variable and . Our equation becomes . By stars and bars, this equation has valid solutions. Now, we use stars and bars on the reds. We must distribute two bars amongst the reds, so we apply stars and bars to . Since each group must have one red, we again do a change of variables with , , and . We are now working on the equation . By stars and bars, this has solutions. The number of valid paths in this case is the number of ways to create the bars times the number of valid arrangements of the stars given fixed bars, which equals . We must multiply by two to account for both cases, so our final answer is .
~ cxsmi
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Fast Video Solution (no stars and bars)
https://youtu.be/BGyRj4yZoAI ~Anton Levonian (Do Math or Go Home)
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.