(a) 单词 REGENERATE 中的 10 个字母共有多少种不同的排列？

(b) 单词 REGENERATE 中的 10 个字母共有多少种不同的排列，要求 4 个 E 全部相邻，且两个 R 之间恰好有 3 个字母？

(c) 从单词 REGENERATE 中 10 个字母的所有排列中随机选出一个，求其中辅音字母（G、N、R、R、T）与元音字母（A、E、E、E、E）交替排列的概率，即任意两个辅音字母不相邻，任意两个元音字母也不相邻。

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。

四E、两R各自相同，其余字母不同。

EEEE四字母块不能放进两R之间仅三格的空隙。因此中间三个字母都从G、N、A、T中有序选，再把R___R和EEEE作两个块。

The EEEE block cannot lie between the Rs, since only three letters are allowed there. Thus the three interior letters are chosen and ordered from G, N, A and T, in $4\times3\times2=4!$ ways.

The R___R block, the EEEE block and the one remaining letter can then be arranged in $3!$ ways. Therefore

先算两种交替模式，每种模式的辅音、元音分别排列，再除以(a)的不同排列总数。

The two possible patterns are CVCVCVCVCV and VCVCVCVCVC.

The five consonants have $\frac{5!}{2!}$ arrangements because the Rs are identical. The five vowels have $\frac{5!}{4!}$ arrangements because the Es are identical. Hence the favourable count is

Using (a),

也可以按随机不放回次序只看辅音元音类别。辅音开头的交替概率与元音开头相同。

Temporarily label repeated letters so that every ordering of the ten individual letters is equally likely. Each unlabelled arrangement corresponds to the same $2!\,4!$ labelled arrangements, so this preserves the required probability.

For the pattern CVCVCVCVCV, the probability is

The other alternating pattern has the same probability. Thus