Math B: Combinatorics and Probability

The combinatorics and probability module counts finite choices and models random outcomes. Define the sample space and repetition rules first; factorials work very hard and will happily work overtime for a bad assumption.

Prerequisites

• Understand sets, functions, and basic proofs. Review the relevant Math A: Discrete Mathematics lessons if needed.
• Be comfortable with algebra and simple Python programs. Probability and number theory start from definitions; expand the first few terms when a summation symbol is unfamiliar.

Coverage

Chapter 6 covers counting and generating functions; Chapter 7 covers graphs, paths, and network flow. Chapter 8 introduces probability and statistical inference, and Chapter 9 develops integers, modular arithmetic, and mathematical questions from cryptography.

Chapter 6

• 6.1 Counting Principles: Addition, Multiplication, Inclusion-Exclusion, and the Pigeonhole Principle
• 6.2 Permutations, Combinations, and Repeated Selection: Do Order and Repetition Matter?
• 6.3 Generating Functions: Encoding Counting Sequences into Algebraic Objects

Chapter 7

• 7.1 Graph Models and Traversal: First Define Edges, Then Choose Representation
• 7.2 Paths, Connectivity, Trees, and Shortest Paths: Different Weights Require Different Algorithms
• 7.3 DAG, Strongly Connected Components and Network Flow: From Dependency Sorting to Resource Allocation

Chapter 8

• 8.1 Probability Space and Conditional Probability: First Build the Model, Then Update Your Judgment
• 8.2 Random Variables, Expectation, and Variance: Beyond the Mean
• 8.3 Sampling and Statistical Inference: From Observed Data to Limited Conclusions

Chapter 9

• 9.1 Divisibility, Congruence, and Euclid's Algorithm: The Fundamental Structure of Integers
• 9.2 Modular Inverses, Fast Exponentiation, and the Chinese Remainder Theorem
• 9.3 Prime Numbers, Factorization, and the Mathematical Boundaries of RSA

Outcomes

• Decide whether order and repetition matter, and check counting formulas with small enumerations.
• Model dependencies, paths, or resource allocation as graphs and choose algorithms suited to their weights and structure.
• State assumptions behind probability models and sampling conclusions, and compute congruences, modular inverses, and fast exponentiation.

How to use this map

Enter at the chapter that fits your problem, checking the required set and proof concepts first. When a counting formula gives a huge number, test it with two or three objects. For probability, write the sample space first. Small-integer number theory experiments explain mathematical relationships; they are not deployable cryptographic systems.