The Selection of Members | 成员的选择

📚 The Selection of Members | 成员的选择

In Edexcel A Level Mathematics, questions on selecting members from a group test your understanding of permutations and combinations. This article explains the key ideas, formulas, and common restrictions step by step.

在爱德思 A Level 数学中，关于从一组中选择成员的问题考查你对排列与组合的理解。本文逐步解释核心概念、公式和常见限制条件。

1. The Fundamental Counting Principle | 基本计数原理

If there are m ways to perform task A and n ways to perform task B independently, then there are m × n ways to perform both tasks in order. For example, choosing a chair from 5 candidates and a secretary from 4 remaining candidates gives 5 × 4 = 20 possible pairs.

若任务 A 有 m 种完成方式，任务 B 有 n 种独立完成方式，则按顺序完成两个任务共有 m × n 种方式。例如，从 5 名候选人中选主席、再从剩余 4 人中选秘书，共有 5 × 4 = 20 种选择。

This principle extends to any number of stages: multiply the number of choices at each stage. It is the foundation of both permutations and combinations.

该原理可扩展到任意多个阶段：将每一阶段的选择数相乘。它是排列与组合的基础。

2. Factorial Notation | 阶乘记号

For a positive integer n, n! = n × (n – 1) × … × 2 × 1. By convention, 0! = 1. Factorials count the number of ways to arrange n distinct objects in a line.

对于正整数 n，n! = n × (n – 1) × … × 2 × 1。按约定，0! = 1。阶乘用于计算 n 个不同对象排成一行的方法数。

You should be comfortable simplifying expressions such as n! / (n – 2)! = n(n – 1), because many exam questions require cancellation before calculation.

你应熟练化简表达式，例如 n! / (n – 2)! = n(n – 1)，因为许多考题要求先约分再计算。

3. Introducing Permutations | 排列介绍

A permutation is an arrangement of objects where order matters. When selecting r members from n distinct members and arranging them, the number of permutations is P(n, r) = n! / (n – r)!.

排列是考虑顺序的对象安排。从 n 个不同成员中选出 r 个并安排顺序，排列数为 P(n, r) = n! / (n – r)!。

For example, the number of ways to choose a chair, vice-chair, and secretary from 8 members is P(8, 3) = 8! / 5! = 8 × 7 × 6 = 336.

例如，从 8 名成员中选出主席、副主席和秘书的方法数为 P(8, 3) = 8! / 5! = 8 × 7 × 6 = 336。

4. Introducing Combinations | 组合介绍

A combination is a selection of objects where order does not matter. The number of ways to select r members from n distinct members is C(n, r) = n! / (r! (n – r)!).

组合是不考虑顺序的对象选择。从 n 个不同成员中选出 r 个的方法数为 C(n, r) = n! / (r! (n – r)!)。

For example, choosing a 3-person committee from 8 members can be done in C(8, 3) = 56 ways, because the order of the three chosen members is irrelevant.

例如，从 8 名成员中选出 3 人委员会有 C(8, 3) = 56 种方式，因为所选 3 人的顺序无关紧要。

5. The Combination Formula nCr | 组合公式 nCr

The formula C(n, r) = n! / (r! (n – r)!) is derived by taking the permutation count P(n, r) = n! / (n – r)! and dividing by r! to remove the ordering of the chosen r members.

公式 C(n, r) = n! / (r! (n – r)!) 的推导方法是：先取排列数 P(n, r) = n! / (n – r)!，再除以 r! 以消除所选 r 个成员的顺序。

Useful identities include C(n, r) = C(n, n – r), and C(n, 0) = C(n, n) = 1. These identities can save time in exams.

常用恒等式包括 C(n, r) = C(n, n – r)，以及 C(n, 0) = C(n, n) = 1。这些恒等式可帮助你在考试中节省时间。

Remember that 0 ≤ r ≤ n must always hold; if r is outside this range, C(n, r) is defined as 0 in most contexts.

记住 0 ≤ r ≤ n 必须始终成立；如果 r 超出此范围，大多数情况下 C(n, r) 被定义为 0。

6. Selecting from Distinct vs Identical Members | 从不同成员与相同成员中选择

In most Edexcel questions, members are distinct individuals, so combinations and permutations apply directly. Be alert for wording like ‘identical’ or ‘indistinguishable’, which changes the counting method.

在大多数爱德思考题中，成员是不同的个体，因此可直接使用组合与排列。注意题目中’相同’或’不可区分’等措辞，这些会改变计数方法。

If some members are identical, you may need to divide by the factorial of the number of identical items, or list possible cases separately. However, A Level questions rarely go beyond simple distinct-member selections.

如果某些成员相同，你可能需要除以相同物品数量的阶乘，或分别列出可能情况。不过 A Level 题目很少超出简单的不同成员选择。

7. Restrictions: Must Include or Exclude | 限制条件：必须包含或排除

When a question says a specific member must be included, first select that member (1 way), then choose the remaining r – 1 from the other n – 1 members: C(n – 1, r – 1).

当题目说明某特定成员必须被选入时，先选择该成员（1 种方式），再从其余 n – 1 名成员中选 r – 1 名：C(n – 1, r – 1)。

If a specific member must be excluded, simply reduce the pool to n – 1 members and select r from them: C(n – 1, r). Treating restrictions first avoids double counting.

如果某特定成员必须被排除，只需将可选范围缩小为 n – 1 名成员，从中选 r 名：C(n – 1, r)。先处理限制条件可避免重复计数。

8. Multi-stage Selections | 多阶段选择

Many exam problems require selecting members from different groups, such as 3 boys from 10 boys and 2 girls from 8 girls. Use the multiplication principle: C(10, 3) × C(8, 2).

许多考题要求从不同组中选择成员，例如从 10 名男生中选 3 名、从 8 名女生中选 2 名。使用乘法原理：C(10, 3) × C(8, 2)。

Be careful: if the problem only specifies the total number of members without group restrictions, you should select from the combined pool, not multiply separate group counts.

注意：如果题目只指定总人数而没有分组限制，则应从合并后的人群中选择，而不是将各组计数相乘。

Always check whether stages are independent; if one choice affects the next, adjust the numbers accordingly.

始终检查各阶段是否独立；如果前一个选择影响后一个选择，要相应地调整数字。

9. Using Selection in Probability | 在概率中应用选择

To find a probability involving a selection, compute the number of favourable outcomes and divide by the total number of possible selections. Both are usually combinations.

要求涉及选择的概率时，计算有利结果数并除以所有可能选择的总数。两者通常都用组合数表示。

For example, if 3 members are chosen at random from 10, the probability that a particular member A is included equals C(1,1) × C(9,2) / C(10,3) = 36 / 120 = 0.3.

例如，从 10 人中随机选 3 人，特定成员 A 被选中的概率为 C(1,1) × C(9,2) / C(10,3) = 36 / 120 = 0.3。

Always ensure the numerator and denominator are counted under the same rules regarding order and restrictions.

始终确保分子与分母在顺序和限制条件方面使用相同的计数规则。

10. Common Exam Mistakes | 常见考试错误

The most common mistake is using permutations when order does not matter, or combinations when order does. Always ask: does swapping two selected members create a different outcome?

最常见的错误是在顺序无关时使用排列，或在顺序有关时使用组合。始终问自己：交换两个所选成员是否会得到不同结果？

Another frequent error is forgetting to subtract excluded members before counting, or multiplying when you should add. Double check whether cases are mutually exclusive or independent.

另一个常见错误是在计数前忘记减去被排除的成员，或在应该相加时错误相乘。反复检查各情况是互斥的还是独立的。

Also watch out for arithmetic slips with factorials; simplify the formula before using a calculator.

还要注意阶乘的算术失误；在使用计算器之前先化简公式。

11. Worked Example: Committee Selection | 例题：委员会选择

Worked example: From 12 people, a committee of 5 is to be chosen. If one particular person must be on the committee, how many committees are possible?

例题：从 12 人中选出一个 5 人委员会。如果某特定人员必须进入委员会，共有多少种可能的委员会？

Solution: First include the required person: 1 way. Choose the remaining 4 from the other 11 people: C(11, 4) = 330. Thus there are 1 × 330 = 330 possible committees.

解：首先纳入该特定人员：1 种方式。从其余 11 人中选 4

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The Selection of Members | 成员的选择