One definition of the factorial that is more general than the usual
is via the gamma function, where
This definition is not limited to positive integers, and in fact can be taken as the definition of the factorial for non-integers. With this definition, you can quite clearly see that
If you are starting from the "usual" definition of the factorial, in my opinion it is best to take the statement 
as a part of the definition of the factorial function, as anything else would require proofs using the factorial to include special cases for 
and 
. It's a definition that is consistent and makes our lives easier.
GeeksforGeeks
geeksforgeeks.org › mathematics › zero-factorial
What is Zero Factorial (0!): Meaning, Calculation, Applications - GeeksforGeeks
December 10, 2023 - The value of Zero factorial is equal to 1. Symbolically, it can be represented as 0! = 1. We can prove that the value of zero factorial is equal to 1 in different ways. As factorial is used to calculate the permutation and combination of any ...
Top answer 1 of 2
4
One definition of the factorial that is more general than the usual
is via the gamma function, where
This definition is not limited to positive integers, and in fact can be taken as the definition of the factorial for non-integers. With this definition, you can quite clearly see that
If you are starting from the "usual" definition of the factorial, in my opinion it is best to take the statement as a part of the definition of the factorial function, as anything else would require proofs using the factorial to include special cases for and . It's a definition that is consistent and makes our lives easier.
2 of 2
1
There is exactly one permutation of a set of zero elements.
Videos
ChiliMath
chilimath.com › home › lessons › intermediate algebra › zero factorial
Zero Factorial | ChiliMath
July 20, 2024 - For the equation to be true, we must force the value of zero factorial to equal 1, and no other. Otherwise, 1!≠1 which is a contradiction. So yes, 0!
Quora
quora.com › Why-is-0-factorial-equal-to-1-Is-there-any-pure-basic-mathematical-proof-mathematics-physics
Why is 0 factorial equal to 1? Is there any pure basic mathematical proof (mathematics, physics)? - Quora
Answer (1 of 7): The simple answer is as follows. n!=(n-1)!n or, turning this round, (n-1)! = n!/n Now put n=1. You get 0!=1!/1 = 1 There is another demonstration, involving integration which involves more advanced maths. I'm not going into it here. Hope this helps.
Reddit
reddit.com › r/math › why the factorial of 0 is always 1?
r/math on Reddit: Why the factorial of 0 is always 1?
January 16, 2022 - If you think of a factorial as a function such that the mth derivative of a nth degree polynomial is ... the correct formula kind pops out at you, but it wouldn't be as obvious if you started from n = 0. It's not a proper proof of why 0!
ThoughtCo
thoughtco.com › why-does-zero-factorial-equal-one-3126598
Why Does Zero Factorial Equal One?
May 18, 2024 - The first reason why zero factorial is equal to one is that the definition of the factorial states that 0! = 1. A definition is a mathematically correct explanation (even if a somewhat unsatisfying one).
The Math Doctors
themathdoctors.org › zero-factorial-why-does-0-1
Zero Factorial: Why Does 0! = 1 ? – The Math Doctors
But that means that x^0 must be 1 because when you multiply x^2 by it, the result is still x^2. Only x^0 = 1 makes sense here. This idea was discussed as part of What Do Exponents Mean?, as well as in Anything to the Zero Power: Why 1? As in many situations involving zero, we ultimately just ...
Top answer 1 of 16
73
We need 
to be defined as 
so that many mathematical formulae work. For example we would like
to work when 
ie 
Also we require that the formula for the number of ways of choosing 
objects from 
is valid for 
ie
is valid when 
Things need to work when we extend our definition of the factorial via the gamma function.
The above gives 
and so we require 
since 
2 of 16
66
I'm not sure that there is anything to prove. I think it follows directly from the definition of factorial:
So if 
the right hand side is the empty product, which is 
by convention.
Homework.Study.com
homework.study.com › explanation › prove-that-0-zero-factorial-is-1.html
Prove that 0! (zero factorial) is 1. | Homework.Study.com
Calculate frac 6 factorial 4 factorial 2 factorial. Prove that division by zero is not allowed (undefined). If n is a positive integer, prove that \int\limits_0^1 (\ln x)^n \text{d}x=( 1)^n n!
Medium
omerseyfeddinkoc.medium.com › why-is-0-1-a-surprisingly-logical-fact-of-mathematics-216b74046d70
Why Is 0! = 1? A Surprisingly Logical Fact of Mathematics | by Omer Seyfeddin Koc | Medium
July 20, 2025 - In fact, the result 0! = 1 is not only mathematically consistent, it’s essential. Let’s unpack this surprising result and explore why it makes perfect sense. The factorial function, denoted with an exclamation point (!), is defined as the product of all positive integers from 1 up to a given number:
Medium
medium.com › math-simplified › why-is-zero-factorial-1-83cb340fe883
Why is Zero Factorial 1?. Beginning with the definition of… | by Safwan | Math Simplified | Medium
October 16, 2021 - Beginning with the definition of factorials we can work our way to a proof where 0! = 1 is mathematically proven. In the field of combinations and permutations, the explanation given is usually that there is only one way to arrange 0 objects, ...
Zero-factorial
zero-factorial.com › whatis.html
0! - What is Zero-Factorial?
Simple answer: 0! (read "Zero Factorial") is defined to equal 1. Involved answer(s): There are several proofs that have been offered to support this common definition.
Reddit
reddit.com › r/explainlikeimfive › eli5: why is the factorial of 0 equal to 1?
r/explainlikeimfive on Reddit: ELI5: Why is the factorial of 0 equal to 1?
October 4, 2023 -
Wouldn't this also mean that 1!=0!, why is this true?
Top answer 1 of 26
640
Others have explained the 0! = "number of ways to rearrange 0 people" part, so I'll try to convince you that 0! shouldn't be 0. You can think of 2 lines of people, with 2 distinct people in first one and 3 in the second. How many distinct way can you rearrange them, assuming that no one leaves their line? Well, as for the first line, there are 2! arrangements. As for the second one, the answer is 3!. Then, since you can pair 1 way of arranging the first line with 1 way of arranging the second line to make 1 distinct arrangement of the 2 lines, the number of distinct arrangements is 2! × 3!. Now, what if the first line is empty? Then the answer should be 0! × 3!. If 0! = 0, this number should be 0. But you should get 3! different arrangements when shuffling the second line, no? Then adding an empty first line shouldn't suddenly make the number of arrangement 0? Look just how convenient it is for the number of permutation of an empty set to be 1!
2 of 26
280
There is an excellent video by Eddie Woo on youtube, I don't know if I can link it but I highly recommend you go watch it. However, the proof basically boils down to: 1! = 1 2! = 2 3! = 6 4! = 24 Find the pattern. To get from 4! to 3!, divide by 4. From 3! to 2!, divide by 3. From 2! to 1!, divide by 2. What will be the next answer? Divide it by 1. 0! = 1. It fits the pattern in our made up language of mathematics.
Stack Exchange
math.stackexchange.com › questions › 1630463 › why-is-0-factorial-1
Why is 0 factorial 1? - Mathematics Stack Exchange
From the permutation formula, we ... eventually be 1/n. Thus, for n!/0! to be equivalent to 1/n, the only logical solution is to allow 0 factorial to be 1....
YouTube
youtube.com › premath
0!=1 PROOF | Zero Factorial is Equal to One - YouTube
Learn the proof to why 0 factorial is equal to 1. Quick and simple explanation by PreMath.com
Published May 24, 2021 Views 25K
Saspublishers
saspublishers.com › article › 2239 › download pdf
Zero Factorial
According to Wikipedia [6], the factorial of a non-negative integer n, denoted by n!, is the product of all · positive integers less than or equal to n. The value of 0! is 1, according to the convention for an empty product.
Math Answers
math.answers.com › other-math › What_is_the_rationale_for_defining_0_factorial_to_be_1
What is the rationale for defining 0 factorial to be 1? - Answers
Only from the definition of a factorial ... n=1 one has (1-1)! = 1!/1 or 0! = 1/1 or 0! = 1. = This is a proof that zero factorial is equal to one to be known....
Lunatic Laboratories
loonylabs.org › 2021 › 02 › 07 › the-zero-factorial
The zero factorial | Lunatic Laboratories
February 7, 2021 - We end up with the 0! in our equation. We know the equation is correct and we know for certain that 1! = 1, for that reason this is proof that ... Now a bit of fine print here… you can actually take the factorial of negative numbers, the math works out, but for practical purposes the n!
CK-12 Foundation
ck12.org › all subjects › cbse math › laws of integral exponents - exponent rules, properties, interactives and examples › why is 0 factorial equal to 1?
Flexi answers - Why is 0 factorial equal to 1? | CK-12 Foundation
September 11, 2025 - The factorial of any positive integer ... positive integers less than or equal to 0. So why is 0! equals to 1? The answer lies in the principle of mathematical induction and the definition of the empty product....