arithmetic
What is a Factorial?
January 17, 2026
5 min read
PanMaths Team
⚡ THE DEFINITION
A factorial is the product of all positive integers from 1 up to a given number n.
n! = n × (n-1) × ... × 1← THE DEFINITION
Introduction to Factorials
In mathematics, the factorial of a positive integer n is the product of all positive integers less than or equal to n. We write this using an exclamation mark: n!. It’s like a descending multiplication chain that always ends at 1.
How to Calculate n!
1
Start with n
Pick your starting integer.
2
Multiply Down
Multiply it by every positive integer smaller than it.
3
Stop at One
The chain always finishes at 1. Never include 0 (or the whole result would be 0!).
Common Factorials
1! = 1Starting value
3! = 63 × 2 × 1 = 6
5! = 1205 × 4 × 3 × 2 × 1 = 120
0! = 1Special case by definition
The Mystery of 0! = 1
It might seem strange that 0! is 1 instead of 0. Here are two reasons why: 1) It represents the number of ways to arrange zero items (there is exactly 1 way: doing nothing!). 2) It ensures that the formula for combinations and permutations works correctly without dividing by zero.
💡Pro Tips for Factorials
- Factorials grow faster than exponential functions (like 2^n).
- You can simplify fractions of factorials readily: 10! / 8! = 10 × 9 = 90.
- They are the secret sauce behind the 'nCr' and 'nPr' buttons on your scientific calculator.
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