(a) 单词 RECORDERS 中的 9 个字母共有多少种不同的排列？

(b) 单词 RECORDERS 中的 9 个字母共有多少种不同的排列，要求开头和末尾各有一个 E，且三个 R 不全部相邻？

将单词 RECORDERS 中的 9 个字母随机分成两组：一组 5 个字母，另一组 4 个字母。

(c) 求三个 R 在同一组的概率。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

三R与两E分别相同，其余不同。

固定两端E，先算所有内部排列，再减去RRR作块的情况。

With an E at each end, the seven interior letters have $\frac{7!}{3!}$ arrangements.

If the Rs are all together, the RRR block and C, O, D, S have $5!$ arrangements. Therefore

也可把内部四个唯一字母先排列，再分三R全部分离或只有一对相邻两类。

Fix the Es at the ends. Arrange C, O, D and S in $4!$ ways, creating five gaps.

• If all Rs are separate, choose three gaps: $\binom53$ choices.
• If exactly two Rs are together, choose a gap for RR and a different gap for the remaining R: $5\times4$ choices.

Thus

随机分组时实际9个字母牌等概率。按三个R落在五字母组或四字母组分类，选择同组其他字母。

There are $\binom95=126$ equally likely choices for the group of five individual letters.

If all Rs are in the group of five, choose two of the six non-R letters in $\binom62=15$ ways. If all Rs are in the group of four, choose its remaining letter in $\binom61=6$ ways; this also uniquely determines the group of five.

Hence

另一条路线把分组看作五个与四个位置，按三个R依次落入哪组计算。

Temporarily label the three Rs and assign all nine individual letters uniformly to five slots in one group and four in the other.

The probability that all Rs occupy the five-slot group is $\frac59\frac48\frac37$, and the probability that all occupy the four-slot group is $\frac49\frac38\frac27$.

These events are disjoint, so