(a) 求单词 HAPPINESS 中 9 个字母的不同排列数。

(b) 求单词 HAPPINESS 中 9 个字母的不同排列数，要求首尾两个字母不相同。

(c) 求单词 HAPPINESS 中 9 个字母的不同排列数，要求两个 P 相邻，且两个 S 之间恰好有两个字母。

将单词 HAPPINESS 中的 9 个字母随机分成一组 5 个和另一组 4 个。

(d) 求两个 P 都在同一组、两个 S 都在另一组的概率。

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。

P和S分别重复两次，除去各自内部排列。

首尾相同只有两P或两S两种情况，从全部排列中减去。

If both ends are P, the seven interior letters have two identical Ss and can be arranged in $\frac{7!}{2!}$ ways. The same count applies when both ends are S. These cases are disjoint, so

也可以按首尾是否含P、S分三类直接相加。

The other five letters H, A, I, N and E are distinct.

Thus

第三种分类固定首字母：若是P或S，末位不能相同；若是另外五个唯一字母，末位自动不同。

If the first letter is P, the remaining P has seven permitted positions, excluding the last position. The other seven positions contain two identical Ss, giving $7\frac{7!}{2!}$ arrangements. The same count applies if the first letter is S.

If the first letter is one of H, A, I, N or E, choose it in 5 ways and arrange the remaining letters in $\frac{8!}{2!\,2!}$ ways.

Hence

按PP是否在两S之间分类。若在其中，只能形成SPPS；否则先有序选两S之间的两个唯一字母，再排两个块与其他字母。

If PP lies between the Ss, it forms the block SPPS. This block and the five distinct letters have $6!$ arrangements.

Otherwise, the two letters between the Ss must be chosen and ordered from H, A, I, N and E, in $5\times4$ ways. The S__S block, the PP block and the remaining three distinct letters then have $5!$ arrangements.

Thus

另一条路线按左S位置分类。固定两S后，数剩余格中能放PP的相邻位置对。

The left S can occupy positions 1 through 6, with the right S three positions later.

If the left S is in position 1 or 6, there are five adjacent pairs of free positions in which PP can be placed. If it is in position 2, 3, 4 or 5, there are four such pairs.

After placing the Ss and PP, the five distinct letters have $5!$ arrangements. Therefore

把9个实际字母随机分组，五个字母组的选择等概率。分五个字母的一组含PP或含SS两种互斥情况。

There are $\binom95$ equally likely choices for the group of five individual letters.

If this group contains both Ps and neither S, choose its other three letters from the five distinct letters, in $\binom53$ ways. If it contains both Ss and neither P, there are another $\binom53$ choices.

Hence

也可把五个字母组与四个字母组看作九个等可能位置，依次放入两个P与两个S。

Temporarily distinguish the two Ps and the two Ss, and assign all letters uniformly to five slots in one group and four in the other.

The probability that both Ps occupy the five-slot group and both Ss the four-slot group is

For the opposite assignment it is

These assignments are disjoint. Hence