Factorial (n!)
n! (“n factorial”) means n × (n − 1) × … × 2 × 1, the product of the whole numbers from 1 to n. So 4! = 4 × 3 × 2 × 1 = 24, and 0! is defined as 1.
How to read it
OpenStax’s Contemporary Mathematics defines n! as the product of every whole number less than or equal to n and works 4! = 24. It is the number of ways to put n different things in order: Oscar Levin’s Discrete Mathematics gives n! as the number of permutations of n elements. The values grow very quickly (Grinstead and Snell, Introduction to Probability), and when one factorial is divided by another, OpenStax cancels the common factors first: 8!/6! = 8 × 7 = 56. Every one of these sources defines 0! as 1; Grinstead and Snell say it is done to make certain formulas come out simpler, Levin “for reasons of convenience”.
Also searched as: 0!, exclamation mark in math, factorial symbol, n!, n factorial, what does ! mean in math.
n!!
Two exclamation marks are a different function, the double factorial. NIST’s Digital Library of Mathematical Functions defines n!! as the product of every other number: 2 × 4 × 6 × … × n for even n, and 1 × 3 × 5 × … × n for odd n.
Read next
- Next in Statistics symbolsCombinations (nCr)nCr is the number of combinations: the number of ways to choose r things from n when the order doesn’t matter. It is nCr = n! / (r!(n − r)!), also written as n over r in tall brackets and read “n choose r”.
- The same shape in Barcodes and QR codesEAN-13 barcodeAn EAN-13 barcode carries a 13-digit product number, the GTIN-13: a GS1 company prefix, the company’s number for the item, and a check digit. It is the shop barcode on most products outside the US and Canada, where the 12-digit UPC is used.
More statistics symbols
The same shape elsewhere
Browse symbols drawn with letters and numbers.
Sources
- Contemporary Mathematics, 7.2 Permutations, OpenStax (Rice University), Factorials; Factorials, Example 7.5
- Introduction to Probability, Charles M. Grinstead and J. Laurie Snell, Introduction to Probability, 2nd edition (2006 CHANCE version), Dartmouth College, Ch. 3.1 Permutations; Factorials; Historical remarks
- Discrete Mathematics, 1.3 Combinations and Permutations, Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd edition, Oscar Levin, University of Northern Colorado (Open Math Books), Permutations
- NIST Digital Library of Mathematical Functions: Introduction (Common Notations and Definitions), U.S. National Institute of Standards and Technology, Common Notations and Definitions
- Earliest Uses of Symbols in Probability and Statistics, Jeff Miller, Earliest Uses of Various Mathematical Symbols, MacTutor History of Mathematics, University of St Andrews, Factorial
Researched, edited and approved by Rory Hansen before publishing. Last reviewed October 9, 2026. How we research
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