(a) 求将单词 ATTENTIVE 的 9 个字母排列成两个 E 相连，但三个 T 不全部相连的不同排列数。

(b) 求将单词 ATTENTIVE 的 9 个字母排列成两个 E 之间至少有四个字母的不同排列数。

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。

先把EE合为一块，排列剩下各字母，再减去TTT也合为一块的排列。

Treat EE as a single block. The eight objects are EE, A, T, T, T, N, I and V, giving $\frac{8!}{3!}$ arrangements.

If the three Ts are also together, treat TTT as a block. There are then six distinct objects, giving $6!$ arrangements.

Thus the required number is

也可以先排EE块和A、N、I、V五个对象，再在六个空隙中放三个T。排除全部T在同一个空隙，只剩三个分开或两个相连的情况。

Arrange EE, A, N, I and V in $5!$ ways, creating six gaps including the end gaps.

For the Ts not all to be together:

• three separate Ts occupy three distinct gaps: $\binom63=20$ choices;
• a pair TT and a single T occupy two different gaps: $6\times5=30$ choices.

Hence

两个E之间可有4、5、6、7个字母。对每一种间隔，数出左E能放的位置；其余7个字母中三个T相同。

If there are $r$ letters between the Es, the left E can be in $8-r$ positions.

Thus the total is

也可以将两E和中间字母作为一个块。先暂时标记三个T，按块内字母数选取和排列，再排列块与余下字母，最后除去标签。

Temporarily label the three Ts, so the seven non-E letters are distinct.

If $r$ letters are between the Es, choose and order them in $\,{}_7\mathrm{P}_r$ ways. The block E…E and the $7-r$ remaining letters can be arranged in $(8-r)!$ ways. Removing the T labels divides by $3!$.

For $r=4,5,6,7$, the total number is