Question

定义一个好词为只由字母 $A$ 、 $B$ 、 $C$ 组成的字母序列，其中某些字母可以不出现，并且 $A$ 后面绝不紧跟 $B$ ， $B$ 后面绝不紧跟 $C$ ， $C$ 后面绝不紧跟 $A$ 。有多少个由七个字母组成的好词？

Define a good word as a sequence of letters that consists only of the letters $A,$ $B,$ and $C$ — some of these letters may not appear in the sequence — and in which $A$ is never immediately followed by $B,$ $B$ is never immediately followed by $C,$ and $C$ is never immediately followed by $A.$ How many seven-letter good words are there?

答案

答案：192

知识点：有限制的排列乘法原理

难度评级：1750

解答：

每个字母都排除恰好一种后继字母（ $A$ 禁止 $B$ ， $B$ 禁止 $C$ ， $C$ 禁止 $A$ ），所以无论刚写下的字母是什么，接下来 $3$ 个字母中恰有 $2$ 个可以使用。

第一个字母有 $3$ 种选择，剩下六个位置每个都有 $2$ 种选择，因此七个字母的好词个数为 $3 \cdot 2^6 = 192$ 。

Each letter rules out exactly one successor ( $A$ forbids $B,$ $B$ forbids $C,$ $C$ forbids $A$ ), so whatever letter has just been written, exactly $2$ of the $3$ letters may come next.

With $3$ choices for the first letter and $2$ for each of the remaining six positions, the number of seven-letter good words is $3 \cdot 2^6 = 192.$

2003 AIME II 第 3 题

答案：192

难度评级：1750

解答：

其他年份的第 3 题