(a) 求将单词 CANADIANS 的 9 个字母排列成三个 A 全部相连、而两个 N 不相邻的不同排列数。

(b) 随机选取单词 CANADIANS 的 9 个字母的一种排列，求两个 N 之间恰有四个字母的概率。

将单词 CANADIANS 的 9 个字母分成三组，分别含 4 个、3 个和 2 个字母。

(c) 若三个 A 必须在同一组，两个 N 必须在同一组，共有多少种分组方式？

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AAA合成一块，先计全部排列，再减去两个N也相邻的排列。两个N相同，计数需要除以2的阶乘。

Treat AAA as one block. Together with C, N, D, I, N and S, there are seven objects, with two identical Ns, giving $\frac{7!}{2!}$ arrangements.

If the Ns are also together, treat NN as one block, giving six distinct objects and $6!$ arrangements. Therefore,

也可以先排AAA块和四个不同字母，再在六个不同空隙中选两个，各放一个N。

Arrange AAA, C, D, I and S in $5!$ ways. These five objects create six gaps, including the end gaps.

Place the two identical Ns in two distinct gaps so that they are not adjacent. The required number is

两个N之间恰有四个字母，N的位置相差5。左N可在前4个位置之一，剩下7格放其他字母，再除以所有不同排列数。

If the left N is in position $i$, the right N must be in position $i+5$. Thus $i=1,2,3,4$ gives four choices.

The remaining seven positions contain three identical As and the four distinct letters C, D, I, S, so the number of favourable arrangements is $4\frac{7!}{3!}=3360$.

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

另一条路线把两N及中间四字母合成一块。暂时标记三个A，先选块内四字母并排序，再排列块和剩余字母，最后消去A的标签。

Temporarily label the three As. Choose and order four of the seven non-N letters to go between the Ns in $\,{}_7\mathrm{P}_4$ ways.

The block N____N and the three remaining labelled letters can be arranged in $4!$ ways. Removing the A labels divides by $3!$. Thus the number of favourable arrangements is

Therefore,

组大小各不相同，所以可直接按4字母组、3字母组、2字母组区分。AAA只能在4字母组或3字母组；NN必须在另一组，逐一分类。

The four distinct letters other than A and N are C, D, I and S. The groups are distinguished by their sizes.

These exhaust the possibilities, since AAA and NN cannot fit in the same group. Therefore the number of divisions is