📚 Combinations: Choosing Without Order | 组合：与顺序无关的选择
Combinations appear throughout A-level Mathematics, especially in probability and statistics. A combination is a selection of objects from a larger set in which the order of the selected objects does not matter. This article builds the core ideas, formula, examples, common errors, and exam strategies step by step.
组合贯穿 A-level 数学，尤其是在概率与统计中。组合是从一个较大的集合中选取若干对象，且所选对象的顺序并不重要。本文逐步构建核心概念、公式、例题、常见错误与考试策略。
1. What Is a Combination? | 什么是组合？
A combination is a way of choosing r objects from n distinct objects without considering the order of selection. For example, selecting a 3-person committee from 8 people is a combination because the committee membership is the same regardless of the order in which the names are read out.
组合是指从 n 个不同对象中选取 r 个对象，而不考虑选取顺序。例如，从 8 人中选出 3 人委员会就是一个组合问题，因为无论名单以何种顺序读出，委员会成员都是相同的。
- Key feature: order does not matter.
- 关键特征：顺序无关紧要。
- Examples: choosing lottery numbers, forming a study group, selecting cards for a hand.
- 示例：选择彩票号码、组成学习小组、选取一手牌。
In set language, a combination is a subset of size r taken from a larger set of size n. The subset {Alice, Bob, Chen} is identical to {Bob, Chen, Alice}.
用集合语言来说，组合是从大小为 n 的较大集合中取出的大小为 r 的子集。子集 {Alice, Bob, Chen} 与 {Bob, Chen, Alice} 完全相同。
2. Why Order Does Not Matter | 为什么顺序不重要
In a combination, the arrangement AB and the arrangement BA represent the same selection. This is the central difference between combinations and permutations. Because order is ignored, each group of r objects is counted only once, not r! times.
在组合中，排列 AB 与排列 BA 代表同一种选择。这是组合与排列的核心区别。由于忽略顺序，每组 r 个对象只被计算一次，而不会被计算 r! 次。
- Choosing 2 letters from A, B, C gives combinations: AB, AC, BC — total 3.
- 从 A、B、C 中选 2 个字母，组合为：AB、AC、BC——共 3 种。
- Arranging 2 letters from A, B, C gives permutations: AB, BA, AC, CA, BC, CB — total 6.
- 从 A、B、C 中排列 2 个字母，排列为：AB、BA、AC、CA、BC、CB——共 6 种。
This example shows that combinations avoid double-counting different orders of the same pair. Therefore, for the same n and r, the number of combinations is always less than or equal to the number of permutations.
这个例子说明组合避免了同一对元素不同顺序的重复计数。因此，对于相同的 n 和 r，组合数总是小于或等于排列数。
3. Combination Notation | 组合的符号
The number of ways to choose r objects from n distinct objects is written as nCr, C(n, r), or as a binomial coefficient. Some calculators label this function as 'nCr'. The standard formula is:
从 n 个不同对象中选取 r 个对象的方法数记作 nCr、C(n, r) 或二项式系数。有些计算器将这一功能标记为 'nCr'。标准公式为：
nCr = n! / (r!(n − r)!)
. When r = 0, choosing no objects is counted as one way, which is consistent with nC0 = 1.
其中 n 是不同对象的总数，r 是选取的数量，且必须满足 n ≥ r ≥ 0。当 r = 0 时，选取 0 个对象被视为一种方式，这与 nC0 = 1 一致。
4. Factorials in the Formula | 公式中的阶乘
A factorial n! is the product of all positive integers from n down to 1. Factorials arise naturally when counting arrangements. The denominator r! in the combination formula removes overcounting caused by arranging the selected r objects in different orders.
阶乘 n! 是从 n 到 1 的所有正整数的乘积。阶乘在计数排列时自然出现。组合公式中的分母 r! 用于消除因将选出的 r 个对象按不同顺序排列而产生的重复计数。
n! = n × (n − 1) × ... × 2 × 1
For example, 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720, and 3! = 6. By definition, 0! ntion is essential because it makes formulas such as nC0 and nCn give the correct value 1.
例如，6! = 6 × 5 × 4 × 3 × 2 × 1 = 720，3! = 6。根据定义，0! = 1。这个约定至关重要，因为它使 nC0 和 nCn 等公式得到正确值 1。
When calculating combinations by hand, it is often easiest to cancel the largest common factorial before multiplying, especially when n is large and r is small.
手工计算组合时，通常最好先约去最大的公同阶乘再相乘，尤其是当 n 较大而 r 较小时。
5. Deriving the Combination Formula | 组合公式的推导
To derive the combination formula, first count permutations of r objects chosen from n objects. This is written as nPr and equals n! / (n − r)!.
要推导组合公式，首先计算从 n 个对象中选取 r 个对象的排列数。该数记作 nPr，等于 n! / (n − r)!。
Each selected set of r objects can be internally arranged in r! different ways. Since a combination ignores internal order, every combination has been counted r! times in the permutation count. Dividing by r! gives:
每组被选出的 r 个对象内部可以有 r! 种不同的排列方式。由于组合忽略内部顺序，每个组合在排列计数中被重复计了 r! 次。除以 r! 得到：
nCr = nPr / r! = n! / (r!(n − r)!)
This derivation shows exactly why r! appears in the denominator and why combinations count subsets rather than ordered lists.
这个推导清楚地说明了为什么分母中会出现 r!，以及为什么组合计算的是子集而不是有序列表。
6. Comparing Permutations and Combinations | 排列与组合对比
The table below summarises the differences between permutations and combinations. Use it as a quick reference when deciding which counting rule applies to a problem.
下表总结了排列与组合的区别。当你判断问题应使用哪种计数规则时，可以将其作为快速参考。
| Feature | Permutation | Published by TutorHao | Exam Prep Revision Series | aleveler.com
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