2008 AMC 8 Problems/Problem 19: Difference between revisions
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<math> \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{2}{7}\qquad\textbf{(C)}\ \frac{4}{11}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{4}{7} </math> | <math> \textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{2}{7}\qquad\textbf{(C)}\ \frac{4}{11}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{4}{7} </math> | ||
==Solution== | ==Solution 1== | ||
The two points are one unit apart at <math>8</math> places around the edge of the square. There are <math> | The two points are one unit apart at <math>8</math> places around the edge of the square. There are <math>8 \choose 2</math><math> = 28</math> ways to choose two points. The probability is | ||
<cmath>\frac{8}{28} = \boxed{\textbf{(B)}\ \frac27}</cmath> | <cmath>\frac{8}{28} = \boxed{\textbf{(B)}\ \frac27}</cmath> | ||
==Solution 2== | |||
Arbitrarily pick a point in the grid. Clearly, we see two options for the other point to be placed, so the answer is <math>\boxed{\textbf{(B)}\ \frac27}</math>. | |||
==Video Solution== | |||
https://www.youtube.com/watch?v=ZVD_PZYjonQ ~David | |||
==Video Solution 2== | |||
https://youtu.be/PeI8YSHCdlM Soo, DRMS, NM | |||
==See Also== | ==See Also== | ||
{{AMC8 box|year=2008|num-b= | {{AMC8 box|year=2008|num-b=18|num-a=20}} | ||
{{MAA Notice}} | |||
Latest revision as of 13:53, 30 December 2024
Problem
Eight points are spaced around at intervals of one unit around a 
square, as shown. Two of the 
points are chosen at random. What is the probability that the two points are one unit apart?
![[asy] size((50)); dot((5,0)); dot((5,5)); dot((0,5)); dot((-5,5)); dot((-5,0)); dot((-5,-5)); dot((0,-5)); dot((5,-5)); [/asy]](http://latex.artofproblemsolving.com/7/b/4/7b4e4d5204c427f5ab2b242c550558317658b12d.png)
Solution 1
The two points are one unit apart at places around the edge of the square. There are 
ways to choose two points. The probability is
![\[\frac{8}{28} = \boxed{\textbf{(B)}\ \frac27}\]](http://latex.artofproblemsolving.com/2/d/8/2d8bb743056c3f8a324def77999d5c702eb2113a.png)
Solution 2
Arbitrarily pick a point in the grid. Clearly, we see two options for the other point to be placed, so the answer is 
.
Video Solution
https://www.youtube.com/watch?v=ZVD_PZYjonQ ~David
Video Solution 2
https://youtu.be/PeI8YSHCdlM Soo, DRMS, NM
See Also
| 2008 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing
