Discrete Mathematics

1. Combinatorics
   1. Basic Principles
      1. Rule of Sum
         1. Conceptual Overview
         2. Applications and Examples
         3. Limitations and Considerations
      2. Rule of Product
         1. Fundamental Concept
         2. Applications in Multi-step Processes
         3. Real-world Examples
   2. Permutations and Combinations
      1. Permutations of Distinct Objects
         1. Definition and Number of Arrangements
         2. Permutations with Restrictions
         3. Cyclic Permutations
         4. Applications in Arranging Items
      2. Permutations with Repetition
         1. Understanding Repeated Elements
         2. Calculation Techniques
         3. Applications in Situations with Identical Items
      3. Circular Permutations
         1. Concept and Differences from Linear Permutations
         2. Calculation Methods
         3. Real-life Examples
      4. Combinations
         1. Definition and Calculation
         2. Combinations with Restrictions
         3. Applications in Selection Problems
      5. Combinations with Repetition
         1. Understanding Multi-Selections
         2. Calculation Techniques
         3. Applications in Resource Allocation
      6. Binomial Coefficients
         1. Definition and Calculation
         2. Properties and Pascal's Identities
         3. Applications in Expanding Polynomials
      7. Pascal's Triangle
         1. Structure and Properties
         2. Applications in Binomial Expansions
         3. Interesting Patterns and Identities
   3. Advanced Counting Techniques
      1. Inclusion-Exclusion Principle
         1. Basic Concept and Formula
         2. Applications in Overlapping Sets
         3. Complex Problem Solving Examples
      2. Pigeonhole Principle
         1. Basic Concept and Statement
         2. Simple and Complex Applications
         3. Theoretical Implications and Problems
      3. Derangement (Permutations with No Fixed Points)
         1. Basic Theory and Formula
         2. Applications in Secret Santa or Matching Problems
         3. Derivation and Examples
      4. Counting with Symmetries
         1. Burnside's Lemma
         2. Polya’s Enumeration Theorem
         3. Applications in Symmetrical Structures
   4. Recurrence Relations
      1. Linear Recurrence Relations
         1. Definition and Examples
         2. Homogeneous and Non-Homogeneous Types
         3. Applications in Algorithm Analysis
      2. Solving Recurrence Relations
         1. Iterative Method
         2. Characteristic Equations
         3. Generating Functions Approach
      3. Applications in Counting Problems
         1. Fibonacci Sequence Derivation
         2. Tower of Hanoi Analysis
         3. Divide-and-Conquer Algorithms
   5. Generating Functions
      1. Definition and Basic Properties
      2. Operations with Generating Functions
      3. Problem Solving using Generating Functions
      4. Applications in Partition Theory and Combinatorial Identities