AMC 8 2018 Problem 11
AMC8 2018 年第 11 题
概率★★★☆☆
Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown. 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?
和其他四个同班同学坐在如下所示的两行三列的座位上准备拍集体照。 如果座位是随机分配的，那么 Abby 和 Bridget 坐在同一行或者同一列的相邻位置上的概率是多少？
- A.
- B.
- C.
- D.
- E.
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
There are a total of 6! ways to arrange the kids. Abby and Bridget can sit adjacent in the same column in 3 ways (the pair occupying column 1, column 2, or column 3). For each of these seat positions, Abby and Bridget can switch seats, and the other 4 people can be arranged in 4! ways, which results in a total of ways to arrange them. By the same logic, there are 4 ways for Abby and Bridget to be placed if they are adjacent in the same row: they can swap seats, and the other 4 people can be arranged in 4! ways, for a total of ways. We sum the 2 possibilities to get , or .
We can ignore other students, and treat Abby and Bridget as indistinguishable (since we only care about adjacency, not their order). Thus, the total number of ways is . In one row, they can be adjacent 2 ways: . In one column, they can only be adjacent 1 way: . Add these cases 4+3=7, and therefore, P(Abby and Bridget sitting adjacent) is .
We can split the seating into two separate cases: if Abby is sitting on the corners, and if Abby isn't. If Abby is sitting in the corners, there is a chance Bridget is sitting next to Abby, so there is a chance for the first case. Meanwhile, if Abby is sitting in the middle row, there is a chance Bridget is sitting next to Abby, so there is a chance for the second case. Therefore, P(Abby and Bridget are sitting adjacent to each other) is , or.
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