(a) 求将单词 ZOOLOGICAL 的 10 个字母排列成三个 O 全部相连、而两个 L 不相邻的不同排列数。

(b) 求将单词 ZOOLOGICAL 的 10 个字母排列成两个 L 之间恰有 5 个字母的不同排列数。

从单词 ZOOLOGICAL 的 10 个字母中随机选出两个字母。

(c) 求这两个字母不同的概率。

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。

将OOO合为一块，先算全部排列，再减去LL也相连的情况。

Treat OOO as a single block. Together with Z, L, G, I, C, A and L, there are eight objects with two identical Ls, giving $\frac{8!}{2!}$ arrangements.

If LL is also a block, there are seven distinct objects, giving $7!$ arrangements. Thus

也可以先排OOO块和五个不同字母，再将两个L放入七个不同空隙，避免相邻。

Arrange OOO, Z, G, I, C and A in $6!$ ways, creating seven gaps including the end gaps.

Choose two distinct gaps for the identical Ls. Hence the required number is

两L之间有5个字母，所以位置相差6。左L只能在前4个位置之一，剩下8格排列其他字母，其中O有3个。

If the left L occupies position $i$, the right L occupies position $i+6$, so $i$ can be 1, 2, 3 or 4.

The eight remaining letters include three identical Os. Thus the required number is

另一条路线把两L及中间五个字母当作一块。暂时区分三个O，选好块内字母并排序，再排列块与余下三字母，最后消去O的标签。

Temporarily label the three Os. Choose and order the five letters between the Ls in $\,{}_8\mathrm{P}_5$ ways.

The block L_____L and the three remaining labelled letters have $4!$ arrangements. Removing the O labels divides by $3!$. Hence

两字母相同只可能是两个O或两个L。按不放回抽取算出这两个概率，再取补集。

The only identical-letter pairs are OO and LL. Thus

也可以用组合数计数。随机选取按10个实物字母处理，全部选法数为45；相同的选法有3种OO和1种LL。

There are $\binom{10}{2}=45$ equally likely selections of two physical letters.

There are $\binom32=3$ OO pairs and $\binom22=1$ LL pair. Hence

直接计数时，按是否取O或L分成四类。除O和L外的五字母互不相同，因此从中任选两个也一定不同。

The five letters other than O and L are Z, G, I, C and A, all distinct.

The different-letter pairs fall into four mutually exclusive cases:

Therefore,

按第一次抽到O、L或其余五个单独字母分类。若第一次是O，剩下9个字母中有7个不是O；若第一次是L，有8个不是L。若第一次是其余单独字母，第二次一定与第一次不同。

Condition on the first letter chosen:

• it is O with probability $\frac3{10}$, after which 7 of the remaining 9 letters are not O;
• it is L with probability $\frac2{10}$, after which 8 of the remaining 9 letters are not L;
• it is one of Z, G, I, C or A with probability $\frac5{10}$. Each occurs only once, so the next letter is certainly different.

Therefore,