2010 AMC 12A Problems/Problem 18
Contents
Problem
A 16-step path is to go from to with each step increasing either the -coordinate or the -coordinate by 1. How many such paths stay outside or on the boundary of the square , at each step?
Solution
Brute Force Solution
The number of ways to reach any point on the grid is equal to the number of ways to reach plus the number of ways to reach . Using this recursion, we can draw the diagram and label each point with the number of ways to reach it and go up until we reach the end. Luckily, the figure is not so big that this is too time-consuming or difficult to do.
For example:
etc.
We soon reach
Combinatorial Solution 1
By symmetry we only need to count the paths that go through the second quadrant (, ).
For each of these paths, let be the first point when it reaches . Clearly and the previous point on such path has to be .
Fix the value of . There are ways how the path can go from to , and then there are ways how the path can go from to .
Hence for we get paths, for we get paths, and for we get paths. This gives us paths through the second quadrant, hence the total number of paths is .
Combinatorial Solution 2
Each path that goes through the second quadrant must pass through exactly one of the points , , and .
There is exactly path of the first kind, paths of the second kind, and paths of the third type. The conclusion remains the same.
See also
2010 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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All AMC 12 Problems and Solutions |
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