Combinatorics

Combinatorics Curriculum

This curriculum provides a structured overview of the study of combinatorics.

Basic Counting Principles

The Fundamental Counting Principle:

• Counting the number of ways to perform a sequence of tasks
• Applications to simple counting problems
• Using tree diagrams to visualize counting problems

Permutations:

• Definition of a permutation
• Permutations of n objects taken r at a time (nPr)
• Permutations with repetition
• Circular permutations
• Applications of permutations

Combinations:

• Definition of a combination
• Combinations of n objects taken r at a time (nCr)
• Combinations with repetition
• Relationship between permutations and combinations
• Applications of combinations

Advanced Topics

The Binomial Theorem:

• Binomial coefficients and Pascal's Triangle
• The Binomial Theorem formula
• Applications of the Binomial Theorem
• Combinatorial proofs of the Binomial Theorem
• Extension to multinomial theorem

Inclusion-Exclusion Principle:

• Counting the number of elements in the union of sets
• The Inclusion-Exclusion Principle for two sets
• The Inclusion-Exclusion Principle for multiple sets
• Applications of the Inclusion-Exclusion Principle
• Derangements

Generating Functions:

• Definition of a generating function
• Ordinary generating functions
• Exponential generating functions
• Using generating functions to solve counting problems
• Applications of generating functions