(a) 将单词 CASABLANCA 中的 10 个字母排列，要求两个 C 不相邻，求不同排列的个数。

(b) 将单词 CASABLANCA 中的 10 个字母排列，要求开头和结尾各为一个 A，且两个 C 之间恰好有 3 个字母，求不同排列的个数。

从单词 CASABLANCA 的 10 个字母中选取 5 个字母。

(c) 若所选的 5 个字母中至少有两个 A 且至多有一个 C，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。

四A、两C重复，先算所有排列，再减去CC作块的情况。

There are $\frac{10!}{4!\,2!}$ unrestricted arrangements. If CC is a block, there are nine objects, including four identical As, giving $\frac{9!}{4!}$ arrangements. Hence

也可以先排八个非C字母，再从九个空隙中选两个放相同的C。

Arrange the eight non-C letters in $\frac{8!}{4!}$ ways. They create nine gaps, from which choose two distinct gaps for the identical Cs. Therefore

固定首尾A，内部八位中两C位置相差4，左C有4种位置；其余六字母有两A重复。

Fix an A at each end. Number the eight interior positions 1 through 8. The Cs occupy positions $(1,5)$, $(2,6)$, $(3,7)$ or $(4,8)$.

For each choice, the remaining letters A, A, S, B, L, N have $\frac{6!}{2!}$ arrangements. Hence

这里数的是不同字母组成，相同A与C不区分。先选A的个数2、3、4，再从C、S、B、L、N五种不同字母选余下字母，自动至多一个C。

Identical letters are not distinguished when counting different selections. With at most one C, the remaining letters are selected from the five distinct types C, S, B, L and N.

• With two As, choose three of these five: $\binom53=10$ selections.
• With three As, choose two: $\binom52=10$ selections.
• With four As, choose one: $\binom51=5$ selections.

Thus