一群朋友由 7 名男性和 4 名女性组成。其中三名男性是兄弟：Ali、Ben 和 Charlie。

(a) 将这 7 名男性排成一列，要求 Ali 和 Ben 不站在彼此旁边。求不同排列的个数。

(b) 将这 7 名男性和 4 名女性排成一列，要求所有男性站在一起，所有女性也站在一起。求不同排列的个数。

(c) 在没有任何限制的情况下，将这 7 名男性和 4 名女性分成一个 6 人组、一个 3 人组和一个 2 人组，共有多少种分法？

(d) 将这 7 名男性和 4 名女性随机分成一个 6 人组、一个 3 人组和一个 2 人组。

求 Ali、Ben 和 Charlie 全部在同一组的概率。

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。

先算7人的全部排列，再减去Ali和Ben作为相邻块的排列，块内有两种次序。

The total number of arrangements is $7!$.

If Ali and Ben are adjacent, treat them as one block. The block and the other five men have $6!$ arrangements, with 2 orders within the block. Thus

也可以先排其他五人，产生六个空隙。Ali和Ben必须占不同空隙，两人有区别，分别选择。

Arrange the five men other than Ali and Ben in $5!$ ways.

They create six gaps including the end gaps. Choose one gap for Ali in 6 ways and a different gap for Ben in 5 ways. Hence

将全体男生和全体女生分别当作一个块，两个块可交换位置，块内再各自排列。

Treat the seven men as one block and the four women as another. The two blocks have $2!$ orders.

The men can be ordered internally in $7!$ ways and the women in $4!$ ways. Hence

三个组大小不同，可依组人数区分。先选6人组，再从剩余5人中选3人，最后2人自动成组。

No division by $3!$ is needed because the groups have different sizes.

三兄弟只能同时在6人组或3人组，不能在2人组。分别计数两种情况，再除以(c)的总分组数。

If the brothers are in the group of 6, choose their three companions from the other eight people, then choose the group of 3 from the five remaining people:

If the brothers form the group of 3, choose the group of 6 from the other eight people:

The two cases are mutually exclusive, so

也可以固定各组对应的不同位置，把三兄弟随机放入11个位置。分别求三人全落在6人组或3人组的概率。

Represent the groups by 6, 3 and 2 distinct slots. Uniformly assigning the people to these slots gives equally likely group divisions, since every division corresponds to $6!\,3!\,2!$ assignments.

The probabilities that all three brothers occupy slots in the group of 6 or the group of 3 are respectively

Thus