(a) 将单词 DELIVERED 中的 9 个字母排列，要求三个 E 全部相邻，且两个 D 不相邻，求不同排列的个数。

(b) 随机选取单词 DELIVERED 的 9 个字母的一种排列，求两个 D 之间恰好有 4 个字母的概率。

从单词 DELIVERED 的 9 个字母中选取 5 个字母。

(c) 若这 5 个字母中至少有一个 D 和至少一个 E，求不同选取方式的种数。

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。

EEE作块后共7个对象含两个D，减去DD也相邻的六对象排列。

Treat EEE as one block. Together with D, D, L, I, V, R, there are $\frac{7!}{2!}$ arrangements.

If DD is also one block, the six distinct objects have $6!$ arrangements. Thus

也可以先排EEE块与L、I、V、R五个不同对象，再从六空隙中选两处放D。

Arrange EEE, L, I, V, R in $5!$ ways, creating six gaps. Place the identical Ds in two distinct gaps. Hence

先暂时标记重复字母，使9!个排列等概率。两D相距5，左D有4个位置，D次序有2种，其余7字母排列。

Temporarily label the two Ds and three Es. Each unlabelled arrangement corresponds to the same $2!\,3!$ labelled arrangements, so this does not change the probability.

Exactly four letters between the Ds means their positions are $(1,6)$, $(2,7)$, $(3,8)$ or $(4,9)$. There are 2 orders for the labelled Ds and $7!$ orders for the remaining letters. Therefore

也可以直接数不同字母排列，每组D位置固定后其余7格包含三枚相同E。

The total number of different arrangements is $\frac{9!}{2!\,3!}=30240$.

For each of the four permitted D-position pairs, the remaining letters have $\frac{7!}{3!}$ arrangements. Hence

数不同字母组成时重复D、E不区分。按D数量1或2及E数量1、2、3列六类，其余从L、I、V、R选择。

Identical Ds and identical Es are not distinguished. The permitted compositions and choices from L, I, V, R are

Therefore