Counting Principles and Binomial Expansion | 计数原理与二项式展开

📚 Counting Principles and Binomial Expansion | 计数原理与二项式展开

Counting principles form the foundation of probability and combinatorics. In IB Mathematics, you are expected to apply the multiplication principle, permutations, and combinations to solve problems, and then connect these ideas to the binomial theorem. This article reviews the essential concepts and provides step-by-step examples aligned with the IB syllabus.

计数原理是概率论与组合数学的基础。在IB数学中，你需要熟练运用乘法原理、排列与组合解决问题，并将这些概念与二项式定理联系起来。本文以IB考纲为指引，系统梳理核心知识并配以例题讲解。

1. Fundamental Counting Principle | 基本计数原理

If one event can occur in m ways and a second independent event can occur in n ways, then the total number of ways both events can occur in sequence is m × n. This is called the multiplication principle.

若一件事有 m 种发生方式，另一件独立事件有 n 种发生方式，则两件事依次发生的总方式数为 m × n，这就是乘法原理。

When choices are made from disjoint sets, the total number of ways to choose one item from any of the sets is the sum of the numbers of choices. This is the addition principle.

当从互不相交的集合中进行选择时，从任一集合中选择一项的总方式数为各集合选择数之和，即加法原理。

Use the multiplication principle when tasks are performed in stages; use the addition principle when tasks are mutually exclusive.

任务分阶段进行时用乘法原理；任务互斥时用加法原理。

2. Permutations | 排列

A permutation is an ordered arrangement of distinct objects. The number of ways to arrange n distinct objects in a row is n! (n factorial).

排列是不同对象的有序安排。将 n 个不同对象排成一行的方式数为 n!（n 的阶乘）。

For selecting and arranging r objects from n distinct objects, the number of permutations is:

从 n 个不同对象中选取并排列 r 个对象，排列数为：

P(n,r) = n! / (n – r)!

For example, P(5,3) = 5 × 4 × 3 =

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