2011 AMC 8 Problems/Problem 12

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

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 $\frac26=\boxed{\textbf{(B)}\ \frac13}$

2011 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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2011 AMC 8 Problems/Problem 12

Solution

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