基本计数原理

考什么 · What it tests

需要先会 · Prerequisites

• 乘法与加法 / Multiplication and addition
• 分类讨论 / Casework
• 集合与 Venn 图入门 / Sets and basic Venn diagrams

常见套路 · Common moves

1. 分步乘、分类加：一件事分几步完成就把每步的选择数相乘，分成几类互不相干就相加——先判断是「步」还是「类」，是一切计数的起点。

   Multiply steps, add cases: if a task is done in stages, multiply the choices at each stage; if it splits into disjoint cases, add them — deciding "stage" vs "case" is where all counting begins.
2. 正难则反用互补：问「至少一个」很难数时，用总数减去「一个都没有」，常一步搞定。

   When the front door is hard, use the complement: for "at least one," subtract "none at all" from the total — often a one-step finish.
3. 有重叠用容斥：两类有交集时 $|A\cup B| = |A| + |B| - |A\cap B|$，别把重叠部分数两遍。

   Overlap calls for inclusion-exclusion: when two sets intersect, $|A\cup B| = |A| + |B| - |A\cap B|$ — don't count the overlap twice.

易错点 · Common pitfalls

• 该乘的时候加、该加的时候乘：分步骤（先选上衣再选裤子）是乘法，出题人常用措辞诱你用错。

  Adding when you should multiply and vice versa: stages (pick a top, then pants) call for multiplication, and the wording is designed to mislead you.
• 「至少」类直接正面硬数，既漏情况又重复——看到「至少」先想互补。

  Counting "at least" head-on, both missing and repeating cases — when you see "at least," reach for the complement first.