Question

Buzz Bunny 在一组台阶上一次一级地上下跳。有多少种方式使 Buzz 从地面开始，跳六次后又回到地面？（例如，一种跳法是上-上-下-下-上-下。）

Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of 6 hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)

$4$

$5$

$6$

$8$

$12$

答案：B

知识点：卡塔兰数系统列举

难度评级：1310

视频讲解：

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文字解答：

所有序列都必须以向上 $(U)$ 开始、以向下 $(D)$ 结束，并且任何时刻向下的次数都不能超过此前向上的次数。可行序列为 $UUUDDD$ $UUDUDD$ $UUDDUD$ $UDUDUD$ $UDUUDD$

共有五种。所以正确答案是 B。

We can deduce from the choices that it is possible to exhaust all possible cases for this problem. Note that all sequences must start with up $(U)$ and end with down $(D)$ , and that it should not be possible to go down more times than Buzz has gone up so far. Keeping this in mind, we can arrive at the following possible cases: $UUUDDD$ $UUDUDD$ $UUDDUD$ $UDUDUD$ $UDUUDD$ which is a total of five possible sequences.

Thus, B is the correct answer.

2024 AMC 8 第 13 题

答案：B

难度评级：1310

视频讲解：

文字解答：

其他年份的第 13 题