题目
(a) Find the number of different arrangements of the 9 letters in the word CROCODILE.
(b) Find the number of different arrangements of the 9 letters in the word CROCODILE in which there is a C at each end and the two Os are not together.
(c) Four letters are selected from the 9 letters in the word CROCODILE.
Find the number of selections in which the number of Cs is not the same as the number of Os.
(d) Find the number of ways in which the 9 letters in the word CROCODILE can be divided into three groups, each containing three letters, if the two Cs must be in different groups.
题目中文翻译
(a) 求单词 CROCODILE 中 9 个字母的不同排列数。
(b) 求单词 CROCODILE 中 9 个字母的不同排列数，要求两端各有一个 C，且两个 O 不相邻。
(c) 从单词 CROCODILE 的 9 个字母中选取 4 个字母。
求所选字母中 C 的个数与 O 的个数不同的选法数。
(d) 将单词 CROCODILE 中的 9 个字母分成三组，每组含三个字母，要求两个 C 必须位于不同的组中。求不同的分组方法数。
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
思路
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两个C与两个O分别重复，除以各自2!。
答题过程
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(b)
解法一
思路
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固定两端C，内部七字母总排列减去OO相邻的排列。
答题过程
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With C fixed at each end, the seven middle letters have arrangements. If the Os are together, regard OO as one object, giving arrangements. Thus
解法二
思路
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先排五个不同的非C、非O字母，再从其六个空隙选两处插O。
答题过程
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Arrange R, D, I, L and E in ways between the two fixed Cs. These five letters create six gaps, including the gaps immediately inside the Cs.
Place one O in each of two different gaps. Therefore
(c)
解法一
思路
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按C与O各有几枚分类；同字母不可再当成不同实体额外相乘。
答题过程
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The five other letters R, D, I, L and E are all distinct. The possible selections are:
| Cs | Os | Other letters | Number |
|---|---|---|---|
Thus