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CIE A-Level Maths Study Notes

4.2.1 Understanding Permutations and Combinations

Permutations and combinations are two fundamental concepts in combinatorics, a branch of mathematics concerned with counting, arrangement, and probability. These concepts are crucial in various fields, including statistics, computer science, and physics.

Permutations (Arrangements)

  • Definition: Ordering of objects.
  • Key Point: Order matters.
  • Example: 'ABC' has permutations like ABC, ACB, BAC, etc.

Combinations (Selections)

  • Definition: Selection of objects regardless of order.
  • Key Point: Order doesn't matter.
  • Example: From 'ABC', combinations of 2 letters are AB, AC, BC.

Permutation Formulas

1. Permutation of n Distinct Objects

  • Formula: P(n)=n!P(n) = n!
  • Use: Arrange n different objects.
  • Example: 4 books arranged in 4!4! ways.

2. Permutation with Repetition

  • Formula: P(n,r)=n!(nr)!P(n, r) = \frac{n!}{(n-r)!}
  • Use: Choose r from n objects.
  • Example: Arrange 3 out of 4 books using P(4,3)P(4, 3) .

Combination Formulas

1. Combination of n Distinct Objects

  • Formula: C(n,r)=n!r!(nr)!C(n, r) = \frac{n!}{r!(n-r)!}
  • Use: Choose r from n objects, order irrelevant.
  • Example: 3 books from 5 using C(5,3)C(5, 3) .


1. Arranging 'MATHS'

  • Steps: Count letters (5). Use 5!5!. Answer: 120 ways.

2. Forming a 3-member Committee from 7

  • Steps: Use C(7,3)=7!3!(73)!C(7, 3) = \frac{7!}{3!(7-3)!} . Answer: 35 ways.

Calculation Approach

  • Identify Problem Type: Permutations (order matters) or combinations (order doesn't matter).
  • Select Formula: Based on problem type.
  • Define Variables: n (total objects), r (objects to arrange/select).
  • Perform Calculations: Apply the formula and calculate.
  • Interpret Results: Understand and explain outcome.

Example: Arranging 5 Books

  • Problem: Find arrangements for 5 distinct books.
  • Formula: Use permutation: P(n)=n!P(n) = n! .
  • Apply: Here, n=5 n = 5 , calculate 5!5!.
  • Calculate: 5!=5×4×3×2×15! = 5 \times 4 \times 3 \times 2 \times 1.
  • Result: 120 different arrangements possible.
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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