题目
The eight digits 1, 2, 2, 3, 4, 4, 4, 5 are arranged in a line.
(a) How many different arrangements are there of these 8 digits?
(b) Find the number of different arrangements of the 8 digits in which there is a 2 at the beginning, a 2 at the end and the three 4s are not all together.
Three digits are selected at random from the eight digits 1, 2, 2, 3, 4, 4, 4, 5.
(c) Find the probability that the three digits are all different.
题目中文翻译
将八个数字 1、2、2、3、4、4、4、5 排成一行。
(a) 这 8 个数字共有多少种不同的排列？
(b) 求这 8 个数字的不同排列数，要求开头和末尾各有一个 2，且三个 4 不全部相邻。
从八个数字 1、2、2、3、4、4、4、5 中随机选取三个数字。
(c) 求这三个数字互不相同的概率。
原卷通用作答要求
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。
解答
(a)
解法一
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两枚2、三枚4分别重复，除去内部排列。
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(b)
解法一
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首尾固定2，内部六位含三枚4。减去三枚4作为一个块的情况。
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With the 2s fixed at the ends, the remaining six digits have arrangements.
If the three 4s are together, treat 444 as one object along with 1, 3 and 5. These four objects have arrangements. Hence
(c)
解法一
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按是否选到一枚2、一枚4分四类，另外三个唯一数字为1、3、5。所有实物数字牌选择等概率。
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There are equally likely selections of three individual digits. To have three different values, select at most one 2 and at most one 4.
| Number of 2s | Number of 4s | Number chosen from 1, 3, 5 | Selections |
|---|---|---|---|
Thus
解法二
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也可以取补集。不合格的选择含一对2、一对4或三个4；含一对4时要求第三枚不是4，三类互斥。
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A selection fails to have three different values in exactly one of these disjoint cases:
- Both 2s, and one of the six non-2 digits: selections.
- Exactly two 4s, and one of the five non-4 digits: selections.
- All three 4s: selection.
No selection of only three digits can contain both a pair of 2s and a pair of 4s. Hence