MathIsimple

Lesson 5-3: Counting Principles

Learn permutations, combinations, and inclusion–exclusion with problem templates, worked examples, and practice.

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Learning Objectives

  • Structure counting with multiplication/addition rules.
  • Compute P(n,k)=n!(n−k)!P(n,k)=\frac{n!}{(n-k)!} and C(n,k)=(nk)C(n,k)=\binom{n}{k}.
  • Apply inclusion–exclusion to avoid double counting.
  • Use templates for typical competition-style problems.

Core Formulas

Permutations/Combinations

P(n,k)=n!(n−k)!P(n,k)=\frac{n!}{(n-k)!}
C(n,k)=(nk)C(n,k)=\binom{n}{k}

Inclusion–Exclusion

∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A\cup B|=|A|+|B|-|A\cap B|
∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣A∩C∣−∣B∩C∣+∣A∩B∩C∣|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|

Worked Examples

Example 1: Unlabeled Pairs

Partition 6 people into 3 unlabeled pairs.

(62)(42)(22)3!=15\frac{\binom{6}{2}\binom{4}{2}\binom{2}{2}}{3!}=15

Example 2: Union Count

Numbers from 1–200 divisible by 2, 3, or 5.

Use inclusion–exclusion to obtain 146.

Practice Problems

Problem 1

How many 5-letter strings from A–Z with no repeated letters?

Problem 2

How many integers from 1–1000 are divisible by 3 or 7?

Key Takeaways

  • Break complex counts into disjoint cases or ordered choices.
  • Use inclusion–exclusion to correct overcounting.
  • Leverage templates for permutations, combinations, and symmetry adjustments.
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