AMC10 2021 A

AMC10 2021 A · Q20

AMC10 2021 A · Q20. It mainly tests Basic counting (rules of product/sum), Casework.

In how many ways can the sequence $1,2,3,4,5$ be rearranged so that no three consecutive terms are increasing and no three consecutive terms are decreasing?

有几种方法可以重新排列序列 $1,2,3,4,5$，使得没有三个连续项递增，也没有三个连续项递减？

(A) 10 10

(B) 18 18

(D) 32 32

(E) 44 44

Answer

Correct choice: (D)

正确答案：（D）

Solution

By symmetry with respect to $3,$ note that $(x_1,x_2,x_3,x_4,x_5)$ is a valid sequence if and only if $(6-x_1,6-x_2,6-x_3,6-x_4,6-x_5)$ is a valid sequence. We enumerate the valid sequences that start with $1,2,31,$ or $32,$ as shown below: There are $16$ valid sequences that start with $1,2,31,$ or $32.$ By symmetry, there are $16$ valid sequences that start with $5,4,35,$ or $34.$ So, the answer is $16+16=\boxed{\textbf{(D)} ~32}.$

由于相对于 $3$ 的对称性，注意到 $(x_1,x_2,x_3,x_4,x_5)$ 是有效序列当且仅当 $(6-x_1,6-x_2,6-x_3,6-x_4,6-x_5)$ 是有效序列。我们枚举以 $1$、$2$、$31$ 或 $32$ 开头的有效序列，如下所示：有 $16$ 个以 $1$、$2$、$31$ 或 $32$ 开头的有效序列。由对称性，有 $16$ 个以 $5$、$4$、$35$ 或 $34$ 开头的有效序列。因此答案是 $16+16=\boxed{\textbf{(D)} ~32}$。

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