Difference between revisions of "2010 AMC 12A Problems/Problem 18"
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== Solution == | == Solution == | ||
− | + | Each path must go through either the second or the fourth quadrant. | |
− | + | Each path that goes through the second quadrant must pass through exactly one of the points <math>(-4,4)</math>, <math>(-3,3)</math>, and <math>(-2,2)</math>. | |
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− | == | + | There are <math>1</math> paths of the first kind, <math>{8\choose 1}^2=64</math> paths of the second kind, and <math>{8\choose 2}^2=28^2=784</math> paths of the third type. |
− | + | Each path that goes through the fourth quadrant must pass through exactly one of the points <math>(4,-4)</math>, <math>(3,-3)</math>, and <math>(2,-2)</math>. | |
− | Each path that goes through the | + | Again, there are <math>1</math> paths of the first kind, <math>{8\choose 1}^2=64</math> paths of the second kind, and <math>{8\choose 2}^2=28^2=784</math> paths of the third type. |
− | + | Hence the total number of paths is <math>2(1+8+784) = 1698</math>. | |
== See also == | == See also == |
Revision as of 16:58, 5 October 2013
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
Each path must go through either the second or the fourth quadrant. Each path that goes through the second quadrant must pass through exactly one of the points , , and .
There are paths of the first kind, paths of the second kind, and paths of the third type. Each path that goes through the fourth quadrant must pass through exactly one of the points , , and . Again, there are paths of the first kind, paths of the second kind, and paths of the third type.
Hence the total number of paths is .
See also
2010 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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