Difference between revisions of "2011 AMC 8 Problems/Problem 12"

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==Solution 1==
 
==Solution 1==
If we designate Angie to be on a certain side, then all placements of the other people can be considered unique. There are then <math>3!=6</math> total seating arrangements. If Carlos is across from Angie, there are only <math>2!=2.</math> ways to fill the remaining two seats. Then the probability Angie and Carlos are seated opposite each other is <math>\frac26=\boxed{\textbf{(B)}\ \frac13}</math>
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If we designate Angie to be on a certain side, then all placements of the other people can be considered unique. There are then <math>3!=6</math> total seating arrangements. If Carlos is across from Angie, there are only <math>2!=2</math> ways to fill the remaining two seats. Then the probability Angie and Carlos are seated opposite each other is <math>\frac26=\boxed{\textbf{(B)}\ \frac13}</math>
  
 
==Solution 2==
 
==Solution 2==

Revision as of 17:53, 27 April 2017

Problem

Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?

$\textbf{(A) } \frac14 \qquad\textbf{(B) } \frac13 \qquad\textbf{(C) } \frac12 \qquad\textbf{(D) } \frac23 \qquad\textbf{(E) } \frac34$

Solution 1

If we designate Angie to be on a certain side, then all placements of the other people can be considered unique. There are then $3!=6$ total seating arrangements. If Carlos is across from Angie, there are only $2!=2$ ways to fill the remaining two seats. Then the probability Angie and Carlos are seated opposite each other is $\frac26=\boxed{\textbf{(B)}\ \frac13}$

Solution 2

If we seat Angie first, there would be only one out of three ways Carlos can sit across from Angie. So the final answer is $\boxed{\textbf{(B) }\frac{1}{3}}$

See Also

2011 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 11
Followed by
Problem 13
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

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