The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n: n! = n \cdot (n-1) \cdots 2 \cdot 1. Factorials appear in permutations, combinations, the binomial theorem, Taylor series, and probability — they are the building block of discrete counting.
Definition
The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n: n! = n \cdot (n-1) \cdots 2 \cdot 1.
💡 Intuition
Factorial counts the number of ways to arrange n distinct objects in a row — for 3 items, there are 3! = 6 possible orderings.
🎯 Core Idea
Factorials grow extremely fast: 10! = 3{,}628{,}800. The key identity is n! = n \cdot (n-1)!, which defines the recursive structure.
Example
Formula
Notation
n! reads 'n factorial'; by convention 0! = 1
🌟 Why It Matters
Factorials appear in permutations, combinations, the binomial theorem, Taylor series, and probability — they are the building block of discrete counting.
💭 Hint When Stuck
Try writing the countdown: 5! = 5 x 4 x 3 x 2 x 1. Start from n and multiply down to 1. Remember 0! = 1 by definition.
Formal View
Related Concepts
🚧 Common Stuck Point
0! = 1 by definition (or convention), not by direct multiplication — this edge case is essential for combinatorial formulas to work when choosing all or no items.
⚠️ Common Mistakes
- Thinking 0! = 0 — by definition, 0! = 1
- Computing n! + m! as (n + m)! — factorials do not distribute over addition
- Underestimating how fast factorials grow — 20! is over 2 \times 10^{18}, far too large for casual computation
Go Deeper
Frequently Asked Questions
What is Factorial in Math?
The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n: n! = n \cdot (n-1) \cdots 2 \cdot 1.
Why is Factorial important?
Factorials appear in permutations, combinations, the binomial theorem, Taylor series, and probability — they are the building block of discrete counting.
What do students usually get wrong about Factorial?
0! = 1 by definition (or convention), not by direct multiplication — this edge case is essential for combinatorial formulas to work when choosing all or no items.
What should I learn before Factorial?
Before studying Factorial, you should understand: multiplication.
Prerequisites
Next Steps
Cross-Subject Connections
How Factorial Connects to Other Ideas
To understand factorial, you should first be comfortable with multiplication. Once you have a solid grasp of factorial, you can move on to permutation and combination.
Visualization
StaticVisual representation of Factorial