What is Combinatorics?

Combinatorics is a branch of mathematics focused on counting, arranging, and selecting objects, often to solve problems involving discrete structures.

Counting Principles

The two fundamental rules are:

• Addition Rule: If there are \(m\) ways to do one thing and \(n\) ways to do another, there are \(m + n\) ways to do one or the other.
• Multiplication Rule: If there are \(m\) ways to do one thing and \(n\) ways to do another, there are \(m \cdot n\) ways to do both.

Example

You have 3 shirts and 2 pants. How many outfits?

Applications

Combinatorics is used in computer science (e.g., algorithms), statistics (e.g., probability), and cryptography (e.g., code design).