Permutations and Combinations

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand the terms permutation and combination, and solve simple problems involving selections
  2. Solve problems about arrangements of objects in a line, including those involving:
    • Repetition (e.g., the number of ways of arranging the letters of the word 'NEEDLESS')
    • Restriction (e.g., the number of ways several people can stand in a line if two particular people must, or must not, stand next to each other)
    • Note: Questions may include cases such as people sitting in two (or more) rows

1. Introduction to Counting: The Basic Multiplication Principle

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When you need to count the number of ways different events can happen, and these events happen one after another, you multiply the number of ways each event can happen.

The Rule: If Event 1 can happen in m ways, and Event 2 can happen in n ways, then both events happening in sequence can occur in m × n ways.

Examples:

Example 1: Choosing Outfits You have 3 shirts and 4 pants. How many different outfits can you make?

Solution:

  • You pick 1 shirt (3 choices)
  • Then you pick 1 pair of pants (4 choices)
  • Total outfits = 3 × 4 = 12 ways

Example 2: Choosing a Meal A restaurant menu has 3 starters, 5 main courses, 4 drinks, and 2 desserts. How many different complete meals can you choose?

Solution:

  • Choose 1 starter (3 ways)
  • Choose 1 main (5 ways)
  • Choose 1 drink (4 ways)
  • Choose 1 dessert (2 ways)
  • Total meals = 3 × 5 × 4 × 2 = 120 ways

Example 3: Answering an MCQ Exam An exam has 5 multiple-choice questions. Each question has 4 options (A, B, C, D). How many different ways can you answer all 5 questions?

Solution:

  • Question 1: 4 choices
  • Question 2: 4 choices
  • Question 3: 4 choices
  • Question 4: 4 choices
  • Question 5: 4 choices
  • Total ways = 4 × 4 × 4 × 4 × 4 = 4⁵ = 1024 ways

Thought Process:

  1. How many events do we have?
  2. How many ways can each event happen?
  3. If events follow each other → MULTIPLY

2. Factorials and Linear Arrangements

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What is a Factorial?

A factorial is a special way of multiplying a number by all the whole numbers below it, down to 1.

Notation: We write "n factorial" as n!

Examples:

  • 3! = 3 × 2 × 1 = 6
  • 4! = 4 × 3 × 2 × 1 = 24
  • 5! = 5 × 4 × 3 × 2 × 1 = 120
  • 1! = 1
  • 0! = 1 (by definition)

When to Use Factorials

Use n! when you need to arrange n different objects in a line, and the number of objects equals the number of spaces available.

Example: 5 Friends Standing in a Line 5 friends (A, B, C, D, and E) stand in a straight line. How many different arrangements are possible?

Solution: Think of it as filling 5 spaces:

  • Space 1: 5 choices (any of the 5 friends)
  • Space 2: 4 choices (one friend already used)
  • Space 3: 3 choices
  • Space 4: 2 choices
  • Space 5: 1 choice

Total = 5 × 4 × 3 × 2 × 1 = 5! = 120 ways

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