Combinatorics and Graph Theory

          Module 1: Fundamentals of Combinatorics

Section 1: Introduction to Counting Principles
- Lesson 1: What is Counting?
  - Module 1, Section 1, Lesson 1 What is Counting?
- Lesson 2: Key Combinatorial Terms
  - Module 1, Section 1, Lesson 2 Key Combinatorial Terms

Section 2: Basic Counting Rules
- Lesson 1: The Addition Principle
  - Module 1, Section 2, Lesson 1 The Addition Principle
- Lesson 2: The Multiplication Principle
  - Module 1, Section 2, Lesson 2 The Multiplication Principle

Section 3: Avoiding Mistakes in Counting
- Lesson 1: Recognizing Overcounting
  - Module 1, Section 3, Lesson 1 Recognizing Overcounting
- Lesson 2: Simple Problems on Arrangements
  - Module 1, Section 3, Lesson 2 Simple Problems on Arrangements

Section 4: Visualizing Counting Problems
- Lesson 1: Using Diagrams to Illustrate Counts
  - Module 1, Section 4, Lesson 1 Using Diagrams to Illustrate Counts
- Lesson 2: Introduction to Venn Diagrams
  - Module 1, Section 4, Lesson 2 Introduction to Venn Diagrams

Section 5: Practice and Review of Basic Concepts
- Lesson 1: Guided Problem Walkthrough
  - Module 1, Section 5, Lesson 1 Guided Problem Walkthrough
- Lesson 2: Review and Practice Exercises
  - Module 1, Section 5, Lesson 2 Review and Practice Exercises

Module 2: Counting Principles in Problem Contexts

Section 1: Real-Life Counting Scenarios
- Lesson 1: Counting in Everyday Situations
  - Module 2, Section 1, Lesson 1 Counting in Everyday Situations
- Lesson 2: Identifying Counting Problems Around Us
  - Module 2, Section 1, Lesson 2 Identifying Counting Problems Around Us

Section 2: Systematic Listing Techniques
- Lesson 1: Organizing Information for Counting
  - Module 2, Section 2, Lesson 1 Organizing Information for Counting
- Lesson 2: Avoiding Duplicate Counts
  - Module 2, Section 2, Lesson 2 Avoiding Duplicate Counts

Section 3: Breaking Down Complex Problems
- Lesson 1: Decomposition Strategies for Counting
  - Module 2, Section 3, Lesson 1 Decomposition Strategies for Counting
- Lesson 2: Step-by-Step Problem Analysis
  - Module 2, Section 3, Lesson 2 Step-by-Step Problem Analysis

Section 4: Introduction to the Pigeonhole Principle
- Lesson 1: Understanding the Concept
  - Module 2, Section 4, Lesson 1 Understanding the Pigeonhole Principle
- Lesson 2: Basic Applications of the Principle
  - Module 2, Section 4, Lesson 2 Basic Applications of the Principle

Section 5: Combining Counting Methods
- Lesson 1: Integrating Multiple Counting Rules
  - Module 2, Section 5, Lesson 1 Integrating Multiple Counting Rules
- Lesson 2: Practice with Combined Techniques
  - Module 2, Section 5, Lesson 2 Practice with Combined Techniques

Module 3: Permutations and Combinations

Section 1: Introduction to Permutations
- Lesson 1: What Are Permutations?
  - Module 3, Section 1, Lesson 1 What Are Permutations?
- Lesson 2: Simple Permutation Problems
  - Module 3, Section 1, Lesson 2 Simple Permutation Problems

Section 2: Introduction to Combinations
- Lesson 1: What Are Combinations?
  - Module 3, Section 2, Lesson 1 What Are Combinations?
- Lesson 2: Solving Basic Combination Problems
  - Module 3, Section 2, Lesson 2 Solving Basic Combination Problems

Section 3: Permutations with Repetition
- Lesson 1: Counting with Repeated Elements
  - Module 3, Section 3, Lesson 1 Counting with Repeated Elements
- Lesson 2: Practice with Repetition Problems
  - Module 3, Section 3, Lesson 2 Practice with Repetition Problems

Section 4: Arrangements and Selections
- Lesson 1: Organizing Groups and Orders
  - Module 3, Section 4, Lesson 1 Organizing Groups and Orders
- Lesson 2: Basic Techniques in Selection
  - Module 3, Section 4, Lesson 2 Basic Techniques in Selection

Section 5: Recap and Integrated Problem Solving
- Lesson 1: Reviewing Permutations and Combinations
  - Module 3, Section 5, Lesson 1 Reviewing Permutations and Combinations
- Lesson 2: Group Problem Solving Session
  - Module 3, Section 5, Lesson 2 Group Problem Solving Session

Module 4: Introduction to Graph Theory

Section 1: Fundamentals of Graph Theory
- Lesson 1: What Is a Graph?
  - Module 4, Section 1, Lesson 1 What Is a Graph?
- Lesson 2: Common Graph Terminology
  - Module 4, Section 1, Lesson 2 Common Graph Terminology

Section 2: Types and Properties of Graphs
- Lesson 1: Simple Graphs and Their Characteristics
  - Module 4, Section 2, Lesson 1 Simple Graphs and Their Characteristics
- Lesson 2: Directed Versus Undirected Graphs
  - Module 4, Section 2, Lesson 2 Directed Versus Undirected Graphs

Section 3: Representing Graphs
- Lesson 1: Drawing Graphs and Diagrams
  - Module 4, Section 3, Lesson 1 Drawing Graphs and Diagrams
- Lesson 2: Using Matrices and Lists
  - Module 4, Section 3, Lesson 2 Using Matrices and Lists

Section 4: Paths, Walks, and Cycles
- Lesson 1: Defining Paths and Walks in Graphs
  - Module 4, Section 4, Lesson 1 Defining Paths and Walks in Graphs
- Lesson 2: Recognizing Simple Cycles
  - Module 4, Section 4, Lesson 2 Recognizing Simple Cycles

Section 5: Basic Graph Problems and Practice
- Lesson 1: Finding Shortest Paths
  - Module 4, Section 5, Lesson 1 Finding Shortest Paths
- Lesson 2: Exploring Connectivity Issues
  - Module 4, Section 5, Lesson 2 Exploring Connectivity Issues

Module 5: Graphs in Combinatorial Problem Solving

Section 1: Graphical Models in Counting
- Lesson 1: Using Graphs to Represent Counting Problems
  - Module 5, Section 1, Lesson 1 Using Graphs to Represent Counting Problems
- Lesson 2: Counting Paths in Graph Structures
  - Module 5, Section 1, Lesson 2 Counting Paths in Graph Structures

Section 2: Trees and Their Properties
- Lesson 1: Introduction to Trees in Graph Theory
  - Module 5, Section 2, Lesson 1 Introduction to Trees in Graph Theory
- Lesson 2: Basic Problems Involving Trees
  - Module 5, Section 2, Lesson 2 Basic Problems Involving Trees

Section 3: Graph Matchings and Pairings
- Lesson 1: What Is a Graph Matching?
  - Module 5, Section 3, Lesson 1 What Is a Graph Matching?
- Lesson 2: Simple Matching Problems
  - Module 5, Section 3, Lesson 2 Simple Matching Problems

Section 4: Practical Applications of Graphs
- Lesson 1: Modeling Networks with Graphs
  - Module 5, Section 4, Lesson 1 Modeling Networks with Graphs
- Lesson 2: Graphs in Real-World Problem Solving
  - Module 5, Section 4, Lesson 2 Graphs in Real-World Problem Solving

Section 5: Integrative Review and Practice
- Lesson 1: Combining Combinatorics and Graph Concepts
  - Module 5, Section 5, Lesson 1 Combining Combinatorics and Graph Concepts
- Lesson 2: Culminating Practice Problems
  - Module 5, Section 5, Lesson 2 Culminating Practice Problems