Question

Alice 拒绝坐在 Bob 或 Carla 旁边。Derek 拒绝坐在 Eric 旁边。在这些条件下，五个人坐成一排 $5$ 把椅子有多少种方式？

Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?

$12$

$16$

$28$

$32$

$40$

答案：C

知识点：有限制的排列分类讨论

难度评级：1660

解答：

若 Alice 坐在一端，她旁边的人不能是 Bob 或 Carla，所以必须是 Derek 或 Eric。

不妨设这个人是 Eric。则 Eric 旁边的人必须是 Bob 或 Carla。之后没有更多限制。

这给出总数 $2 \cdot 2 \cdot 2 \cdot 2 = 16.$

第一个 $2$ 表示两个端点。第二个 $2$ 表示 Derek 或 Eric。第三个 $2$ 表示 Bob 或 Carla。最后一个 $2$ 表示剩下的 $2$ 个人。

若 Alice 坐在中间三个位置之一，则她两侧必须是 Derek 和 Eric；Bob 和 Carla 只能坐在最后 $2$ 个座位。

Alice 的位置有 $3$ 种，Derek 和 Eric 左右交换有 $2$ 种，Bob 和 Carla 填剩下两个座位有 $2$ 种。

这一类共有种。 $3 \cdot 2 \cdot 2 = 12$

总数为 $12 + 16 = 28$ 。

所以正确答案是 C。

If Alice sits on an end, then the person next to her cannot be Bob or Carla. This means that it must be Derek or Eric.

WLOG, let the person be Eric. Then the person next to Eric has to be Bob or Carla. After that there are no more restrictions.

This gives us a total of $2 \cdot 2 \cdot 2 \cdot 2 = 16.$

The first $2$ is for both edges. The second $2$ is for Derek or Eric. The third $2$ is for Bob or Carla. The final $2$ is just for the $2$ people that are remaining.

Otherwise, let Alice be in one of the three non-end seats. Then the two people next to her have to be Derek and Eric. Bob and Carla are forced to be in the last $2$ seats.

There are $3$ choices for Alice's seat. The side on which Derek sits has $2$ options, and then there are $2$ options for where Bob and Carla go.

This gives us $3 \cdot 2 \cdot 2 = 12$ configurations.

Therefore, there are a total of $12 + 16 = 28$ total seating arrangements.

Thus, C is the correct answer.

2017 AMC 10A 第 19 题

答案：C

难度评级：1660

解答：

其他年份的第 19 题