Table of Contents
What Is a Permutation?
A permutation is an arrangement of objects in a specific order. Unlike combinations, permutations consider the sequence of selection. For example, the arrangements ABC and BAC are considered different permutations but the same combination.
Permutations are fundamental in probability theory, cryptography, and combinatorics. They help answer questions like "How many ways can you arrange 3 books on a shelf from 10 books?" or "How many different PIN codes are possible?"
Permutation Formula
Where n is the total number of items and r is the number of items being chosen. The exclamation mark denotes the factorial function.
Common Permutation Values
| n | r | P(n,r) | Example |
|---|---|---|---|
| 5 | 2 | 20 | 2-letter codes from 5 letters |
| 10 | 3 | 720 | 3-digit codes from 10 digits |
| 26 | 3 | 15,600 | 3-letter codes from alphabet |
| 52 | 5 | 311,875,200 | 5-card sequences from deck |
Permutations vs Combinations
- Permutations: Order matters. ABC ≠ BAC. Use when arrangement or sequence is important.
- Combinations: Order does not matter. {A,B,C} = {B,A,C}. Use when only selection matters.
- P(n,r) is always ≥ C(n,r) because P(n,r) = C(n,r) × r!
Frequently Asked Questions
What is 0 factorial?
By definition, 0! = 1. This is a mathematical convention that makes many formulas work correctly, including the permutation formula when r = n (all items are arranged).
When do I use permutations in real life?
Permutations apply when order matters: assigning race positions, creating passwords, scheduling tasks, arranging seats, or any scenario where different sequences produce different outcomes.
What is the maximum n value?
Most calculators can handle n up to about 170, because 171! exceeds the maximum floating-point number. For larger values, logarithmic or arbitrary-precision methods are needed.