Counting Principles Summary

Combinatorics · Axiom Academy

Let's review the six fundamental counting principles that form the foundation of combinatorics. Core Idea: If choices are mutually exclusive (cannot happen together), add the possibilities Formula: If A and B are disjoint sets, then |A ∪ B| = |A| + |B| When to Use: "Either/or" scenarios where options don't overlap Example: Menu has 5 appetizers or 8 desserts - choose one item: 5 + 8 = 13 choices Core Idea: If making sequential independent choices, multiply the possibilities Formula: If task has m ways for first step and n ways for second step, total ways = m × n When to Use: "And then" scenarios where you make multiple choices in sequence Example: 3 shirts and 4 pants: 3 × 4 = 12 outfits Core Idea: Count everything, then subtract what you don't want Formula: |A - B| = |A| - |A ∩ B| or use complement: |A'| = |U| - |A| When to Use: "At least one" problems or when easier to count what's excluded Example: 5-letter words without vowels: 26⁵ - (words with at least one vowel) Core Idea: If arrangements are overcounted by a factor, divide to remove duplicates Formula: Distinct arrangements = (Total arrangements) / (Symmetry factor) When to Use: When order doesn't matter or objects are indistinguishable Example: Seating 3 people at a round table: 3! / 3 = 2 unique arrangements Core Idea: Count overlapping sets by adding individuals, subtracting overlaps, adding back triple overlaps Formula: |A ∪ B| = |A| + |B| - |A ∩ B|

This is the written version of the interactive lesson above. See the full Combinatorics course.