is a mathematical operation denoted by an exclamation mark (!) that represents the product of all positive integers from 1 up to a given number. For a non-negative integer $ n $, the factorial $ n! $ is calculated as:
For example:
$ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 $
$ 3! = 3 \times 2 \times 1 = 6 $
Special case: The factorial of 0 is defined as $ 0! = 1 $, based on the convention of the empty product.
Factorials are widely used in combinatorics, probability, and permutations, where they count the number of ways to arrange $ n $ distinct objects. For instance, $ 5! $ gives the number of ways to order 5 unique items.
The factorial function can also be defined recursively:
Note: Factorials are only defined for non-negative integers. Negative numbers do not have factorials.
product of all integers between 1 and the integral input of the function
Wikipedia
en.wikipedia.org › wiki › Factorial
Factorial - Wikipedia
1 week ago - Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book Sefer Yetzirah. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of
Reddit
reddit.com › r/explainlikeimfive › eli5: what is a factorial and how does it work
r/explainlikeimfive on Reddit: ELI5: What is a factorial and how does it work
December 31, 2024 - A way to understand what a factorial represents is that it's the number of ways you can arrange X items. for example if you had 6 teddy bears, and you lined them up, you'd have 6 choices for the first bear, 5 choices for the second bear, and so on.
ELI5: Factorials
You're counting how many ways you can arrange things.
Say you have 4 cards. How many possible ways can you order those cards?
There are 4 possibilities for the first card, 3 for the second, 2 for the third, and one for the last (whichever one is left). 4 * 3 * 2 * 1 = 4!.
More on reddit.com 6
4
August 25, 2011Factorials - Explain like im a two year old
The notation "1*2*...*(n-1)*n" simply means to multiply all the integers from 1 to n. If n = 1, then n-1 is not included since it is not between 1 and n. You may be more satisfied with a recursive definition: Define 0! = 1 Define n! = n*(n-1)! for any positive integer n. More on reddit.com
25
63
May 12, 2019What does factorial mean realistically ?
It is describing how many ways that a substitution key can be created. Walk through it this way. Assume that we start with A. What alphabet replaces 'A'. It could be anything from A to Z. So there are 26 choices of a substitute. OK. Let's for the purpose of this example say 'A's will be replaced by 'K'. Now go to B. What alphabet replaces B? Since we already assigned K to replace A, there are only 25 alphabets to choose from. By the time you get to Z, there will be only one unassigned substitute. So how many keys total? There are 26 choices from our starting point, then 25, then 24 all the way to 1. Therefore there are 26 * 25 * 24 *... * 1 ways of creating a unique key and this is called 26! from the definition of factorials. More on reddit.com
23
124
June 12, 2020Factorials
Maybe they do it with logs. log (n!)1/n = (1/n ) * sum(log(1),...,log(n)). If you have log tables this is a lot easier than the original computation.
I don't know if computers do this anymore. This is what they did before there was electricity, let alone Wolfram Alpha. It's amazing to think of the mathematicians and physicists in those days, calculating everything by hand.
More on reddit.com 8
7
July 21, 2011What is the meaning of 5 factorial?
The meaning of 5 factorial is that we need to multiply the numbers from 1 to 5. That means, 5! = 5 × 4 × 3 × 2 × 1 = 120.
byjus.com
byjus.com › maths › factorial
Factorial
What is the symbol of factorial?
The factorial function is a mathematical formula represented by an exclamation mark “!”. For example, the factorial of 8 can be represented as 8! and it is read as eight factorial.
byjus.com
byjus.com › maths › factorial
Factorial
How do you calculate a factorial?
A factorial is calculated by starting with the number and multiplying it by each previous integer until reaching 1. The factorial is represented by an exclamation point. x! = x * (x-1) * (x-2) ...1. For example, 4! = 4 * 3 * 2 * 1 =24.
study.com
study.com › courses › math courses › math 101: college algebra
Factorial | Definition, Examples & Operations - Lesson | Study.com
Videos
Purplemath
purplemath.com › modules › factorial.htm
What are factorials, and how do they work? | Purplemath
The factorial of a whole number n, denoted as n!, is the product of all the whole numbers between 1 and n: 1×2×3×…×(n−1)×n. So 3! would be 1×2×3 = 6.
BYJUS
byjus.com › maths › factorial
Factorial
In short, a factorial is a function that multiplies a number by every number below it till 1. For example, the factorial of 3 represents the multiplication of numbers 3, 2, 1, i.e. 3! = 3 × 2 × 1 and is equal to 6.
Published October 6, 2021 Views 1K
Study.com
study.com › courses › math courses › math 101: college algebra
Factorial | Definition, Examples & Operations - Lesson | Study.com
July 9, 2012 - How do factorials work? A factorial is an extension of the fundamental counting principle. This principle states that if one thing can be done m ways and another can be done n ways, then both things can occur (m * n ) ways.
YouTube
youtube.com › shorts › R0FwN1BvFUI
Factorials? 5! 🤔 What is a Factorial? - YouTube
What does 5! mean in math? Learn the concept of factorials in less than 60 seconds!This video is AI generated.#Factorial #MathShorts #LearnMath #Shorts #Math...
Published August 8, 2025
Math is Fun
mathsisfun.com › numbers › factorial.html
Factorial Function !
The factorial function (symbol: !) says to multiply all whole numbers from our chosen number down to 1. Examples:
Wolfram MathWorld
mathworld.wolfram.com › Factorial.html
Factorial -- from Wolfram MathWorld
September 13, 2025 - The factorial n! is defined for a positive integer n as n!=n(n-1)...2·1. (1) So, for example, 4!=4·3·2·1=24. The notation n! was introduced by Christian Kramp (Kramp 1808; Cajori 1993, p. 72).
Cuemath
cuemath.com › numbers › factorial
Factorial - Meaning, Formula | Factorial of Hundred & 0
The factorial of a whole number is the function that multiplies the number by every natural number below it. Symbolically, a factorial can be represented by using the symbol "!". This symbol lies on the same key above "1" on a computer keyboard.
Encyclopedia Britannica
britannica.com › science › mathematics
Factorial | Definition, Symbol, & Facts | Britannica
1 week ago - Factorial, in mathematics, the product of all positive integers less than or equal to a given positive integer and denoted by that integer and an exclamation point. Thus, factorial seven is written 7!, meaning 1 × 2 × 3 × 4 × 5 × 6 × 7. ...
freeCodeCamp
freecodecamp.org › news › what-is-a-factorial
What is a Factorial? How to Calculate Factorials with Examples
August 3, 2022 - A factorial is a mathematical operation that you write like this: n!. It represents the multiplication of all numbers between 1 and n. So if you were to have 3!, for example, you'd compute 3 x 2 x 1 (which = 6). Let's see how it works with some more
Penn State Statistics
online.stat.psu.edu › statprogram › reviews › algebra › factorials
A.3 Factorials | STAT ONLINE
A factorial is a mathematical operation in which you multiply the given number by all of the positive whole numbers less than it. In other words \(n!=n \times (n-1) \times …
Factorial US
factorialhr.com › number function factorial
The Factorial Function
The factorial function is a mathematical formula represented by an exclamation mark "!".
YouTube
youtube.com › watch
What is a factorial - YouTube
👉 Learn all about factorials. Factorials are the multiplication of a number in descending integer values back to one. Factorials are used often in sequenc...
Published April 22, 2013
Fonzi AI
fonzi.ai › blog › what-is-a-factorial
What Is a Factorial? Simple Definition + Examples - Fonzi AI Recruiter
August 14, 2025 - Defined for non-negative integers, the factorial function represents the product of all positive integers up to a given integer n. It is denoted by the exclamation mark (!), indicating multiplication of all positive integers up to that number.
Edumentors
edumentors.co.uk › answer › maths › gcse › what-is-a-factorial-and-why-is-it-useful-in-mathematics
What is a factorial and why is it useful in mathematics?
A factorial is the product of an integer and all the integers below it. It's denoted with the symbol !. For example, 5! = 54321 = 120. Factorials are
Quality Gurus
qualitygurus.com › what-is-a-factorial
What is a Factorial? | Quality Gurus
April 24, 2023 - Factorial value is a mathematical ... The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers up to and including n.Factorials are denoted by...
Python
docs.python.org › 3 › library › math.html
math — Mathematical functions
1 week ago - Also called the binomial coefficient because it is equivalent to the coefficient of k-th term in polynomial expansion of (1 + x)ⁿ. Raises TypeError if either of the arguments are not integers. Raises ValueError if either of the arguments are negative. Added in version 3.8. ... Return factorial of the nonnegative integer n.
YouTube
youtube.com › watch
Factorials Explained - YouTube
Factorials Explained: in this video we explain what a factorial is, how to calculate them and how to use the factorial formula. Practice questions and answer...
Published August 8, 2023
ThoughtCo
thoughtco.com › why-does-zero-factorial-equal-one-3126598
Why Does Zero Factorial Equal One?
May 18, 2024 - Still, one must remember that a factorial is the product of all integers equal to or less than the original number—in other words, a factorial is the number of combinations possible with numbers less than or equal to that number.