Mathematical Induction 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.

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Enrolment valid for 12 months

This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.

[NUC Core] MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the mathematics every scientist, engineer, and computer scientist must know. This track covers the NUC Core syllabus for MTH 101: Elementary Mathematics I. From set theory and sequences to trigonometry and complex numbers, you will gain the exact tools used in higher mathematics, algorithm design, data analysis, physics, and engineering. This programme is for first-year university students in mathematics, computer science, engineering, or physics, as well as serious learners preparing for undergraduate study. It also serves professionals who need a rigorous mathematical foundation for technical fields. By the end, you will handle formal set notation, progressions, quadratic models, combinatorics, induction proofs, binomial expansions, trigonometric analysis, and complex arithmetic. You will be equipped to solve real problems in science, engineering, computing, and finance, and ready for advanced study in calculus, statistics, and applied mathematics.

Course Chapters

1. Introduction

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.

Chapter lessons

1-1. Welcome

A direct statement of the course's purpose and structure. This lesson explains how mathematical induction, combinatorics, and the binomial theorem provide essential tools for proof, counting, and algebraic expansion.

1-2. The logic of induction

Explains the underlying logic of mathematical induction. It uses the domino analogy to illustrate how proving a base case and an inductive step proves a statement for all natural numbers.

1-3. The fundamental counting principle

Introduces the multiplication principle as the basis for solving all counting problems. It establishes how to find the total number of outcomes for a multi-stage process.

1-4. Factorial notation

Defines factorial notation (n!) as the product of all positive integers up to n. This is a critical prerequisite for permutations, combinations, and the binomial theorem.

2. Applying Mathematical Induction

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.

Chapter lessons

2-1. The two-step process

Formally outlines the two required steps for a valid proof by induction: establishing the base case, and proving the inductive step by assuming the k-th case is true.

3. Permutations and Combinations

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.

Chapter lessons

3-1. Permutations

Defines permutations as ordered arrangements of objects. This lesson covers the notation and formula for calculating the number of possible permutations.

3-2. Combinations

Defines combinations as unordered selections of objects. The lesson clarifies the difference from permutations and presents the standard formula for calculating combinations.

4. The Binomial Theorem

This chapter provides a complete treatment of the binomial theorem. It presents the formula for expanding binomial expressions of the form (a+b)ⁿ 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.

Chapter lessons

4-1. Statement of the theorem

Formally states the binomial theorem. It connects the coefficients of the expansion to the combination formula from the previous chapter.

4-2. Pascal's triangle

Introduces Pascal's triangle as a recursive method for finding binomial coefficients. This lesson shows its construction and direct relationship to binomial expansions.

5. Conclusion

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.

Chapter lessons

5-1. Summary of concepts

A concise review of the key methods and formulas covered in the course. This lesson ensures the principles of induction, combinatorics, and the binomial theorem are consolidated.

5-2. Applications in probability and statistics

Explains how combinatorics is the foundation of probability theory and how the binomial theorem is directly linked to the binomial distribution in statistics.