Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2008 AMC 8 Problems/Problem 19"

(Created page with "==Problem 19== Eight points are spaced around at intervals of one unit around a <math>2 \times 2</math> square, as shown. Two of the <math>8</math> points are chosen at random. W...")
 
 
(11 intermediate revisions by 10 users not shown)
Line 1: Line 1:
−
==Problem 19==
+
==Problem==
 
Eight points are spaced around at intervals of one unit around a <math>2 \times 2</math> square, as shown. Two of the <math>8</math> points are chosen at random. What is the probability that the two points are one unit apart?
 
Eight points are spaced around at intervals of one unit around a <math>2 \times 2</math> square, as shown. Two of the <math>8</math> points are chosen at random. What is the probability that the two points are one unit apart?
 
<asy>
 
<asy>
Line 13: Line 13:
 
</asy>
 
</asy>
 
<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 1==
 +
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>
 +
 +
==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=21|num-a=20}}
+
{{AMC8 box|year=2008|num-b=18|num-a=20}}
 +
{{MAA Notice}}

Latest revision as of 23:41, 1 July 2023

Problem

Eight points are spaced around at intervals of one unit around a $2 \times 2$ square, as shown. Two of the $8$ 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] $\textbf{(A)}\ \frac{1}{4}\qquad\textbf{(B)}\ \frac{2}{7}\qquad\textbf{(C)}\ \frac{4}{11}\qquad\textbf{(D)}\ \frac{1}{2}\qquad\textbf{(E)}\ \frac{4}{7}$

Solution 1

The two points are one unit apart at $8$ places around the edge of the square. There are $8 \choose 2$$= 28$ ways to choose two points. The probability is

\[\frac{8}{28} = \boxed{\textbf{(B)}\ \frac27}\]

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 $\boxed{\textbf{(B)}\ \frac27}$


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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2008_AMC_8_Problems/Problem_19&oldid=195059"