Permutation and Combination - Mathematics (Undergraduate Foundation)

Stop counting one by one. This course provides the mathematical tools to calculate arrangements and selections with speed. You will master the fundamental counting rules including product, addition, subtraction, and division. The curriculum covers linear and cyclic permutations, identical objects, and restricted placements where items must be together or apart. We conclude with combinations, derangements, and the calculation of shapes within geometric frameworks. Counting is the engine of probability and modern computing. These skills are vital for cryptography, software development, and statistical analysis. You will apply these methods to calculate network paths, secure passwords, and optimise logistics. In any technical career, the ability to quantify possibilities is the difference between guessing and knowing. You will solve complex arrangement problems using standard formulas and logical shortcuts. You will handle specific constraints like fixed positions or items that cannot be next to each other. You will learn to distinguish between permutations and combinations in practical scenarios. You will also calculate the number of lines, triangles, and diagonals in geometric figures using combinatorial principles. This foundation is for undergraduate students and secondary school leavers aiming for high scores in competitive exams. It is also essential for professionals in engineering, finance, and data science who require a refresher on discrete mathematics. Anyone seeking to improve their logical deduction and analytical speed will find value here.

12 hrs

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.

MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the foundational mathematical structures essential for success in quantitative undergraduate degrees and professional technical roles. This comprehensive learning track delivers a rigorous treatment of algebra and trigonometry, moving rapidly from fundamental set theory and real number operations to advanced topics including matrix algebra, complex numbers, and analytical trigonometry. You will establish the critical problem-solving framework required for advanced study in calculus, engineering mechanics, and data science. This programme is primarily designed for first-year university students in STEM disciplines requiring strong analytical bases, particularly engineering, physics, computer science, and economics. It also serves as an intensive, high-level refresher for professionals returning to academia or shifting into data-driven roles demanding precise numerical literacy and logical structuring. Prior competence in standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of core definitions. Upon completion, you will possess the skills to construct rigorous logical arguments using set theory and mathematical induction, model complex relationships with functions and matrices, and analyze periodic systems using advanced trigonometry. You will demonstrate competence in solving diverse equation types, from quadratics to linear systems, and manipulating complex numbers in engineering applications. This track prepares you directly for the mathematical demands of second-year university studies and technical professional certification exams.

Course Chapters

1. Introduction

This chapter introduces the logic of counting required for all arrangement and selection problems. You will master the foundational rules used to calculate total outcomes without manual listing, forming the essential groundwork for the entire course. You will understand and apply the product rule for sequential events; the sum rule for separate choices; the difference rule for total minus unwanted cases; and the division rule to remove duplicate counts.

Concept Overviews

5 Lessons

1:45:53

Problem Walkthroughs

4 Lessons

45:42

2. Permutation

Permutation focuses on the arrangement of objects where the specific order is essential. You will learn to calculate total outcomes when selecting and ordering items from a set, a skill required for creating secure PINs and scheduling tasks. This chapter provides the mathematical formulas needed to handle arrangements of all available items or just a specific portion of them. You will master the use of factorials to count arrangements of all n objects; apply the nPr formula to arrange r items from a larger set; and solve practical problems involving distinct objects in fixed positions.

Concept Overviews

2 Lessons

36:38

Problem Walkthroughs

2 Lessons

11:45

3. Cyclic Permutation

Cyclic permutation covers arrangements in a circle where the starting position is not fixed. You will learn to remove repeated patterns to ensure accurate counting. This section explains the difference between arrangements viewed from one side and those that can be turned over. You will master the formula for circular arrangements where clockwise and anticlockwise orders differ; calculate unique patterns for items that can be flipped like beads; and solve problems involving symmetry.

Concept Overviews

1 Lesson

22:38

Problem Walkthroughs

2 Lessons

8:36

4. Permutation with Identical Objects

Permutation with identical objects covers arrangements where some items are not distinguishable. Swapping identical items creates no new pattern, so you must divide to find unique results. This is essential for calculating arrangements in words, passwords, and digital data accurately. You will master the formula for n items with repetitions; calculate arrangements for words with repeated letters; solve grid-based path problems; and use factorial division to remove duplicate counts.

Concept Overviews

1 Lesson

23:40

Problem Walkthroughs

2 Lessons

11:02

5. Conditional Permutation

Conditional permutation deals with arrangements limited by specific rules. You will learn to count outcomes when items must stay together or stay apart. This logic is essential for advanced scheduling and security where specific conditions apply. You will master the tie-together method for items that stay together; apply the gap method for items that must stay apart; solve problems with fixed positions; and calculate arrangements with restricted letter and digit placements.

Concept Overviews

3 Lessons

39:45

Problem Walkthroughs

3 Lessons

44:06

6. Combination

Combination focuses on selection where order does not matter. It is essential for picking committees or groups where picking A then B is the same as B then A. You will learn to identify selection tasks to avoid overcounting and ensure accurate results. You will master the nCr formula; solve committee problems with group restrictions; calculate lines and triangles in geometry; and handle selections from multiple sets.

Concept Overviews

2 Lessons

55:38

Problem Walkthroughs

3 Lessons

19:32

7. Derangements

Derangements cover scenarios where no item in a set returns to its original position. This chapter explains how to count failed matches, such as putting letters into wrong envelopes or seating people in different chairs. Mastering this prevents overcounting and solves problems where specific items must not occupy their natural spots. You will learn the derangement formula, calculate subfactorials for small sets, solve problems with restricted positions, and apply these rules to real-world logic puzzles.

Concept Overviews

2 Lessons

1:25:08

Problem Walkthroughs

2 Lessons

18:24

8. Geometric Applications

This chapter applies combinatorial logic to solve geometry problems involving points, lines, and shapes. You will learn to calculate the number of possible geometric figures that can be formed from a given set of points, which is essential for spatial reasoning and advanced coordinate geometry. By the end of this chapter, you will master calculating the number of straight lines from points; determining the number of triangles and quadrilaterals; and finding the total diagonals in a polygon.

Concept Overviews

1 Lesson

10:49

Problem Walkthroughs

3 Lessons

27:28