(a) 求将单词 DAFFODILS 的 9 个字母排列成两端各有一个 D，且两个 F 不相邻的不同排列数。

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

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。

固定首尾的D以后，内部7个字母中两个F相同。先算全部排列，再减去FF相邻的排列。

Fix a D at each end. The remaining seven letters are A, F, F, O, I, L, S, giving $\frac{7!}{2!}$ arrangements.

If FF is treated as one block, there are six distinct objects, giving $6!$ arrangements. Hence

也可以先排五个不同字母A、O、I、L、S，再把两个F放在六个不同空隙中。

Arrange A, O, I, L, S in $5!$ ways between the fixed Ds. These letters create six gaps, all between the two Ds.

Choose two distinct gaps and place one F in each. Thus the required number is

两个D之间恰有4个字母，位置相差5。左D有4种位置，剩下7格含两个相同的F；最后除以全部不同排列数。

The possible position pairs for the Ds are $(1,6),(2,7),(3,8),(4,9)$.

For each pair, the other seven letters have $\frac{7!}{2!}$ arrangements, so the number of favourable arrangements is

The total number of different arrangements is $\frac{9!}{2!\,2!}=90720$. Therefore,

另一条路线按夹在D之间的F数分类，将D及中间四字母合成块。0、1、2个F的三种情况需要分别处理相同F的排列。

The five letters other than D and F are A, O, I, L, S.

Treat the string D____D as one block. Count according to the number of Fs inside it:

• No F inside: choose and order four of the five distinct letters, then arrange the block, the remaining distinct letter and two identical Fs:

• One F inside: choose three of the five distinct letters and arrange them with F. The block and the three remaining distinct letters have $4!$ arrangements:

• Two Fs inside: choose two of the five distinct letters, arrange the four internal letters accounting for the identical Fs, then arrange the block and the three remaining letters:

Therefore,