---
title: Permutations
description: Permutations are the distinct arrangements of a set of items where the order of selection is significant. Learners will understand how to calculate permutations, analyze symmetric groups, and solve ordering problems in probability.
category: mathematics
subcategory: combinatorics
difficulty: beginner, intermediate, advanced
url: /subject/permutations
---

# Permutations

Permutations are the distinct arrangements of a set of items where the order of selection is significant. Learners will understand how to calculate permutations, analyze symmetric groups, and solve ordering problems in probability.

## Available Resources

2 Books • 1 Courses • 4 Websites

## Websites

### 1. Brilliant.org - Permutations

Brilliant's interactive, problem-first lessons on counting arrangements: the rule of product, factorials, permutations with repetition and restriction, and the step to combinations. Most content needs a paid subscription. Learners build intuition by solving short puzzles rather than reading proofs.

**Difficulty:** Beginner | **Price:** Paid

**Link:** https://brilliant.org/courses/permutations-and-combinations/

**Tags:** permutations, combinatorics, counting-principles, factorials, interactive-practice

### 2. Khan Academy - Permutations

Khan Academy video lesson from its precalculus probability and combinatorics unit, deriving the permutation formula by counting ordered arrangements. Followed by practice exercises, it prepares learners to count arrangements, compute n!/(n−k)!, and tell when order matters.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:prob-comb/x9e81a4f98389efdf:permutations/v/permutations

**Tags:** permutations, factorials, counting-principles, combinatorics, precalculus

### 3. Art of Problem Solving - Combinatorics

The Art of Problem Solving wiki's combinatorics hub, linking concise articles on counting principles, permutations, combinations, the binomial theorem, stars and bars, and inclusion-exclusion, with competition problems. Suited to students preparing for contests such as AMC and AIME.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://artofproblemsolving.com/wiki/index.php/Combinatorics

**Tags:** combinatorics, combinations, counting, competition-math, binomial-theorem

### 4. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Mathematics for Computer Science (MIT 6.042J)

**Author:** Tom Leighton, Marten van Dijk

Discrete mathematics for computer science with an emphasis on definitions and proofs: logic, induction, sets and relations, graph theory, modular arithmetic, asymptotics, counting and discrete probability. 25 lecture videos, problem sets and exams with solutions build fluency in writing proofs.

**Difficulty:** Intermediate | **Price:** Free

**Link:** https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-fall-2010/

**Tags:** discrete-mathematics, mathematical-proofs, induction, graph-theory, combinatorics, discrete-probability

## Books

### 1. Concrete Mathematics

**Author:** Ronald L. Graham, Donald E. Knuth, Oren Patashnik

Graham, Knuth and Patashnik's Stanford-course text on the discrete mathematics behind algorithm analysis: sums, recurrences, binomial coefficients, special numbers, generating functions and asymptotics. Readers learn to evaluate sums and solve counting problems in closed form, aided by hundreds of graded exercises with answers.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0201142368?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, combinatorics

### 2. Enumerative Combinatorics, Volume 1

**Author:** Richard P. Stanley

Graduate-level reference on counting: sieve methods, partially ordered sets and Möbius inversion, rational generating functions, and permutation statistics such as descents and inversions. Its exercises, each rated by difficulty and with solutions, train readers to find bijective and generating-function proofs.

**Difficulty:** Intermediate | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/0521663512?tag=edmonddante07-20

**Tags:** books, mathematics-statistics, combinatorics

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