Permutation, Combination and the Binomial Theorem - Mathematics (Undergraduate Foundation)

This course covers two critical topics in discrete mathematics: the principle of mathematical induction as a formal proof technique, and the binomial theorem for expanding powers of binomials. The course structure directly links the proof method to one of its most important applications in algebra. These topics are the foundation of algorithm verification, probability theory, and statistics. Mathematical induction is a core technique for proving algorithm correctness and establishing formulas for series. The binomial theorem is a powerful tool used directly in statistics for the binomial distribution and in financial mathematics. By the end of this course, you will be able to apply the principle of mathematical induction to prove mathematical statements, solve complex counting problems using permutations and combinations, and use the binomial theorem to expand any binomial expression for a positive integer exponent. This course is for first-year university students and is a fundamental prerequisite for computer science, statistics, and probability theory. It provides essential skills for competitive mathematics and advanced algebraic studies.

Enrolment valid for 12 months

Course Chapters

1. Introduction
4
This chapter establishes the essential groundwork for the course. It introduces the core logical principles and tools???induction, counting, and factorials???that are required to master the more advanced topics that follow. Key concepts covered are the logic of inductive reasoning, the fundamental principle for counting outcomes, and the definition and use of factorial notation.
Concept Overviews
4 Lessons
2. Applying Mathematical Induction
1
4
This chapter moves from the logic of induction to its formal application. It details the rigorous two-step process for constructing a valid inductive proof, a core skill in formal mathematics. Key topics include the structure of an inductive proof. Worked examples demonstrate the application to summation formulas, divisibility statements, and inequalities.
Concept Overviews
1 Lesson
Problem Walkthroughs
4 Lessons
3. Permutations and Combinations
2
4
This chapter covers combinatorics, the mathematics of counting. A mastery of this topic is the absolute prerequisite for understanding the coefficients in the binomial theorem and for any study of probability. Key topics include the distinction between permutations (arrangements) and combinations (selections), their respective formulas, and their application to problem-solving.
Concept Overviews
2 Lessons
Problem Walkthroughs
4 Lessons
4. The Binomial Theorem
2
4
This chapter provides a complete treatment of the binomial theorem. It presents the formula for expanding binomial expressions of the form (a+b)??? for any positive integer n, a fundamental tool in algebra and statistics. Key topics include the statement of the theorem, the role of Pascal's triangle in finding coefficients, and methods for finding a specific term or coefficient in an expansion.
Concept Overviews
2 Lessons
Problem Walkthroughs
4 Lessons
5. Conclusion
2
This chapter consolidates the concepts of mathematical induction and the binomial theorem. It summarises the proof technique and the expansion formula, reinforcing their connection and importance. The conclusion summarises the two-step process of induction and the binomial formula. It also outlines how these concepts are directly applied in probability and statistics.
Concept Overviews
2 Lessons