What does factorial mean?
In mathematics, a factorial is the product obtained by multiplying a positive whole number by every positive whole number beneath it down to 1. It is written with an exclamation mark: 5 factorial, or 5!, equals 5 × 4 × 3 × 2 × 1, which is 120. Factorials answer one of the most natural counting questions imaginable — how many different ways can a set of things be ordered? — so they lie at the heart of permutations, combinations, and probability. They also grow with astonishing speed: while 10! is just 3,628,800, even modest inputs quickly produce figures larger than the number of atoms in the universe. Beyond pure mathematics, the word appears in statistics in phrases such as "factorial design," describing experiments that vary multiple factors at once. Compact to write and easy to define yet explosive in behavior, the factorial is a small piece of notation with enormous consequences.
The product of all positive integers from 1 up to a given positive integer n, written n!.
"Since 4! = 24, there are twenty-four distinct ways to order four items."
By convention, 0! is defined as equal to 1.
Relating to, involving, or of the nature of factors or factorials.
"The study employed a factorial design varying both dosage and frequency."
The plural is rare but legitimate when referring to multiple factorial values collectively.
"The table lists the first ten factorials."
The exclamation mark in math isn't excitement — it means 'multiply everything below me,' and factorials grow so fast that 70! already exceeds the number of atoms estimated to exist in the observable universe.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of factorial
The word "factorial" comes from "factor," which derives from Latin "factor" meaning "a maker or doer," from the verb "facere" ("to do" or "to make"). The mathematical term arose in the early nineteenth century, when mathematicians needed a name for the product of a descending sequence of factors; the now-standard notation using an exclamation mark was introduced by Christian Kramp around 1808. Related English words sharing the same Latin root include "factory," "manufacture," and "factitious."
Related word forms
How factorial is actually used
"Factorial" is primarily a technical term used in mathematics, statistics, and combinatorics; outside these contexts it appears mainly in its adjectival sense in phrases like "factorial design." The noun takes no plural in normal mathematical use, since each factorial is a single value.
Easily confused with factorial
A factor is a number that divides another exactly, whereas a factorial is the product of a whole series of successive integers.