AMC8 2011

AMC8 2011 · Q12

AMC8 2011 · Q12. It mainly tests Basic counting (rules of product/sum), Probability (basic).

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?

Angie、Bridget、Carlos和Diego随机围坐在一张方桌旁，每边一人。Angie和Carlos坐在对面的概率是多少？

(A) $\frac{1}{4}$ $\frac{1}{4}$

(B) $\frac{1}{3}$ $\frac{1}{3}$

(D) $\frac{2}{3}$ $\frac{2}{3}$

(E) $\frac{3}{4}$ $\frac{3}{4}$

Answer

Correct choice: (B)

正确答案：（B）

Solution

If we designate a person to be on a certain side, then all placements of the other people can be considered unique. WLOG, assign Angie to be on the side. 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 $\frac26=\boxed{\textbf{(B)}\ \frac13}$ .

如果我们指定一个人坐在某一边，那么其他人的所有排列都可以视为唯一的。不失一般性，将Angie指定坐在一边。那么总共有$3!=6$种座位安排。如果Carlos坐在Angie对面，那么剩余两个座位有$2!=2$种填充方式。那么Angie和Carlos坐在对面的概率是$\frac26=\boxed{\textbf{(B)}\ \frac13}$。

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