AMC 8 2026 Problem 17
AMC8 2026 年第 17 题
排列与组合★★★☆☆
Four students are seated in a row. They chat with the people sitting next to them, then rearrange themselves so that they are no longer seated next to any of the same people. How many rearrangements are possible?
四名学生坐在一排。他们先和相邻的人聊天，然后重新排列座位，使得每个人都不再与之前相邻的人坐在一起。有多少种这样的重新排列方式？
- A.2
- B.4
- C.9
- D.12
- E.24
正确答案 · Correct Answer
A
解析 · Solution
解法1
原顺序为 ABCD。重排后 A 不能与 B 相邻，B 不能与 C 相邻，C 不能与 D 相邻。满足条件的排列只有 CADB 和 BDAC 两种。答案：2。
B 原来与 A 和 C 都相邻，重排后 B 只能与 D 相邻，所以 B 必须在两端。同理 C 也必须在另一端。B 和 C 占据两端有 2 种方式，中间 A 和 D 随之固定。答案：2。
Solution 1
Original order: ABCD. After rearrangement, A cannot be next to B, B cannot be next to C, C cannot be next to D. The only valid orders are CADB and BDAC. Answer: 2.
B originally sits next to both A and C, so after rearrangement B can only sit next to D. Similarly, C can only sit next to A. This forces B and C to be at the two ends. There are 2! = 2 ways to place them at the ends, and the middle two positions are then fixed. Answer: 2.
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