Discrete Math

Study of mathematical structures that are fundamentally discrete rather than continuous.

Set TheoryLogic and ProofFunctions and Relations
Graph TheoryCombinatoricsNumber Theory
Cryptography and SecurityNetwork AnalysisScheduling and Planning
Breaking Down ProblemsUsing Visuals and Diagrams
Set TheoryLogic and ProofFunctions and Relati...Graph TheoryCombinatoricsNumber TheoryCryptography and Sec...Network AnalysisScheduling and Plann...Breaking Down Proble...Using Visuals and Di...
Advanced Topics

Combinatorics

Counting Possibilities

Combinatorics is the art of counting how many ways things can happen. It includes concepts like permutations (ordering) and combinations (choosing groups).

Key Ideas

  • Permutation: Arranging objects in order. The number of ways to arrange \(n\) objects is \(n!\) (factorial).
  • Combination: Selecting objects without caring about order. The number of ways to choose \(r\) objects from \(n\) is given by \( \binom{n}{r} \).

Why Is It Useful?

Combinatorics helps answer questions like "How many different pizzas can I make with 5 toppings?" or "How many passwords are possible with 8 letters?"

Real-World Uses

From designing lottery systems to optimizing schedules and analyzing data, combinatorics is everywhere!

Key Formula

\[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]

Examples

  • Counting the number of ways to choose 3 books from a shelf of 10.

  • Calculating how many different ways you can arrange 5 colored beads.

In a Nutshell

Combinatorics is about counting and arranging, helping us find possibilities in many scenarios.

Key Terms

Permutation
An arrangement of objects in a specific order.
Combination
A selection of objects where order doesn't matter.
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Number Theory