AMC8 2018

AMC8 2018 · Q11

AMC8 2018 · Q11. It mainly tests Basic counting (rules of product/sum), Probability (basic).

Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown. X X X X X X If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column?

Abby、Bridget 和他们的四位同学将坐在两排三列的位置上拍团体照，如图所示。 X X X X X X 如果座位位置是随机分配的，那么Abby和Bridget在同一行或同一列相邻的概率是多少？

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

(B) $\frac{2}{5}$ $\frac{2}{5}$

(D) $\frac{1}{2}$ $\frac{1}{2}$

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

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): There are $6$ possible positions for Abby, and this leaves $5$ possible positions for Bridget. Because their order doesn’t matter, the two girls can be placed in any of $\frac{6\cdot5}{2}=15$ pairs of positions. There are $2$ pairs of positions that are adjacent in the top row, $2$ pairs that are adjacent in the bottom row, and $3$ pairs that are adjacent in the same column. So the probability that they occupy adjacent positions is $\frac{2+2+3}{15}=\frac{7}{15}$.

答案（C）：Abby 有 $6$ 个可能的位置，这样 Bridget 剩下 $5$ 个可能的位置。由于两人的先后顺序不重要，这两个女孩可以放在 $\frac{6\cdot5}{2}=15$ 对位置中的任意一对。上排有 $2$ 对相邻位置，下排有 $2$ 对相邻位置，同一列有 $3$ 对相邻位置。因此，她们占据相邻位置的概率是 $\frac{2+2+3}{15}=\frac{7}{15}$。

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