Probability Theory

1. Combinatorics and Probability
   1. Fundamental Counting Principle
      1. Definition and Explanation
      2. Application in Simple Scenarios
      3. Application in Complex Scenarios
      4. Use in Problem Solving
   2. Permutations
      1. Definition of Permutations
      2. Permutations of Distinct Objects
         1. Formula for Permutations
         2. Examples and Illustrations
      3. Permutations with Repetition
         1. Calculating Permutations with Repeated Elements
         2. Applications and Examples
      4. Circular Permutations
         1. Understanding Circular Arrangements
         2. Formula for Circular Permutations
         3. Application in Real-life Situations
      5. Permutations of Multisets
         1. Definition and Explanation
         2. Use of Multinomial Coefficients
         3. Practical Examples
      6. Advanced Counting Techniques
         1. Inclusion-Exclusion Principle
         2. Applications in Permutations
         3. Problem Solving Techniques
   3. Combinations
      1. Definition of Combinations
      2. Combinations of Distinct Objects
         1. Formula for Combinations
         2. Examples and Problem Solving
      3. Combinations with Repetition
         1. Explanation and Formula
         2. Examples in Practical Situations
      4. Binomial Theorem
         1. Statement and Explanation
         2. Expanding Binomials Using Combinations
         3. Applications in Probability
      5. Pascal’s Triangle
         1. Structure and Properties
         2. Relationship to Binomial Coefficients
         3. Problem Solving
   4. Advanced Topics in Combinatorics
      1. Pigeonhole Principle
         1. Basic Explanation and Applications
         2. Advanced Problems and Solutions
      2. Combinatorial Identities
         1. Important Identities and Formulas
         2. Proof Techniques
         3. Applications in Probability
      3. Generating Functions
         1. Definition and Construction
         2. Use in Counting Problems
         3. Solving Recurrence Relations
      4. Catalan Numbers
         1. Definition and Enumeration Problems
         2. Applications in Combinatorial Structures
         3. Related Sequence and Patterns
   5. Applications of Combinatorics in Probability
      1. Probability in Card Games
         1. Calculating Hands and Probabilities
         2. Application of Combinatorics in Strategy
      2. Lottery Probability
         1. Combinatorics in Lottery Outcomes
         2. Estimation of Winning Odds
      3. Combinatorial Probability
         1. Basic Problems and Examples
         2. Conditional Probability with Combinations
      4. Cryptography
         1. Use of Combinatorics in Cryptographic Algorithms
         2. Analysis of Security Based on Combinatorial Problems
      5. Network Theory
         1. Applications in Communication Networks
         2. Graph Theoretical Concepts