CountingThe essence is 'enumerating all possible sequence numbers under a given scheme.' It is the cornerstone of combinatorics, aiming to break down complex overall events into simpler, mutually exclusive or ordered local operations through logical decomposition.
Imagine you want to enter a courtyard:
1. Addition Principle (Parallel): There are $m$ gates on the left side of the wall and $n$ gates on the right. You only need to choose one to enter. Number of methods: $m + n$.
2. Multiplication Principle (Sequential): You must pass through two walls. The first wall has $m$ gates, and the second has $n$ gates. You must complete both stages. Number of methods: $m \times n$.
Imagine you want to enter a courtyard:
1. Addition Principle (Parallel): There are $m$ gates on the left side of the wall and $n$ gates on the right. You only need to choose one to enter. Number of methods: $m + n$.
2. Multiplication Principle (Sequential): You must pass through two walls. The first wall has $m$ gates, and the second has $n$ gates. You must complete both stages. Number of methods: $m \times n$.
Core Definitions of Counting Principles
Addition Principle for Classification: For 'classification' problems, each category can independently complete the task. The core is the disjoint union of sets: $N = m_1 + m_2 + \dots + m_n$.
Multiplication Principle for Sequential Steps: For 'sequential steps' problems, each step depends on the previous one, and all steps must be completed to finish the task. The core is the Cartesian product of paths: $N = m_1 \times m_2 \times \dots \times m_n$.
Classification emphasizes 'independence and completeness'; sequential steps emphasize 'dependency and continuity'.
Independent classification $\implies \sum n_i$
Sequential steps $\implies \prod m_i$
Sequential steps $\implies \prod m_i$