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! Answer from lIllIIIIIlI on reddit.com
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!
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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.
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).
Videos
02:17
Why is 0! equal to 1? - YouTube
Zero factorial or 0! (video) | Probability
04:16
Intuiting Why Zero Factorial Is One // Math Mini - YouTube
r/math on Reddit: Why 0! Is 1...I’m sure you have all seen this ...
03:30
0!=1 PROOF | Zero Factorial is Equal to One - YouTube
01:58
Prove that Zero Factorial is Equal to One - YouTube
The Math Doctors
themathdoctors.org › zero-factorial-why-does-0-1
Zero Factorial: Why Does 0! = 1 ? – The Math Doctors
Recall that “factorial” means the product of descending integers, and is written as \(n!=n(n-1)(n-2)\dots1\). For example, \(5!=5\cdot4\cdot3\cdot2\cdot1=120\). Since we are multiplying factors from n down to 1, it makes sense (mostly) that \(1!=1\), since we just start and end with 1; but how can you count down from 0 to 1??
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 - ... The standard definition seems ... equivalent. In the standard definition 0! is explicitly defined. 0! = 1 because it's defined that way....
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 - One of the strongest arguments for defining 0! = 1 is that it makes other mathematical formulas work seamlessly. ... If we set n=k=0, we’re asking: “In how many ways can we choose 0 items from a set of 0 items?” The answer should be 1, ...
Top answer 1 of 5
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Mostly it is based on convention, when one wants to define the quantity 
for example. An intuitive way to look at it is 
counts the number of ways to arrange 
distinct objects in a line, and there is only one way to arrange nothing.
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10
In a combinatorial sense, 
refers to the number of ways of permuting 
objects. There is exactly one way to permute 0 objects, that is doing nothing, so 
.
There are plenty of resources that already answer this question. Also see:
Link
http://wiki.answers.com/Q/Why_is_zero_factorial_equal_to_one
http://en.wikipedia.org/wiki/Factorial#Definition
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!
Medium
ali.medium.com › why-does-zero-factorial-0-equal-one-82a948c7c95b
Why Does Zero Factorial (0!) Equal One? | by Ali | Medium
September 5, 2024 - As I wandered among terms such as “electricity,” “soccer,” “spacecraft,” and “iPhone” in my head, I suddenly came across what deserved the distinction to be coined as “humanity’s greatest discovery.” In the words of the beloved late artist Bob Ross, “let’s draw a happy tree here” and similarly “let’s draw a circle here and call it zero” to illustrate Al-Khwarizmi’s effort to define nothingness 1200 years ago, which is arguable, in my opinion, humanity’s most remarkable discovery.
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.
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1
There is exactly one permutation of a set of zero elements.
Wikipedia
en.wikipedia.org › wiki › Factorial
Factorial - Wikipedia
1 week ago - {\displaystyle 0!=\Gamma (0+1)=1} , and the gamma function is defined as a continuous function of complex numbers that does not involve a separate choice at this value. The earliest uses of the factorial function involve counting permutations: there are
Math Answers
math.answers.com › other-math › Why_factorial_of_0_equals_1
Why factorial of 0 equals 1? - Answers
The simple answer is that it is defined to be 1. But there is reason behind the decision.As you know, the factorial of a number (n) is equal to:n! = n * (n-1) * (n-2) ... * 1Another way of writing this is:n! = n * (n-1)!Suppose n=1:1! = 1 * ...
GeeksforGeeks
geeksforgeeks.org › mathematics › zero-factorial
What is Zero Factorial (0!): Meaning, Calculation, Applications - GeeksforGeeks
December 7, 2023 - The value of Zero factorial is ... and combination of any number, logically the meaning of zero factorial is to arrange data that contains no value....
Quora
quora.com › Why-does-zero-factorial-0-equal-one-1-1
Why does zero factorial (0!) equal one (1)? - Quora
Answer (1 of 553): Mathematically, n! = n * (n-1) * (n-2) * . . . . . . * 3 * 2 * 1 Dividing by n both sides, n! / n = (n-1) * (n-2) * . . . . . . * 3 * 2 * 1 n! / n = (n-1)! Hence, we have concluded that, (n-1)! = n! / n Now putting n = 1 in the above derived equation, (1 - 1)! = 1! / 1 ...
GeeksforGeeks
geeksforgeeks.org › dsa › why-factorial-of-0-is-1
GFact | Why Factorial of 0 is 1? - GeeksforGeeks
October 4, 2023 - The value of LHS should be equal ... recursive definition of factorial, we see that 0! must be 1 to maintain consistency with the factorial function's definition....
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.
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....
ScienceABC
scienceabc.com › home › pure sciences › why is the factorial of zero equal to one?
Zero Factorial: Why Is The Value Of Zero Factorial Equal To One?
October 19, 2023 - Or, 0! = 1 · However, this explanation tells us nothing about why factorials of negative numbers cannot exist. Let’s turn towards our pattern again to find out why. . . 2!/2 = 1! 1!/1 = 0! 0!/0 = …uh-oh · I’d agree that these methods are a little fishy; they are seemingly sly, implicit ways to determine the factorial of zero.
Dhgate
smart.dhgate.com › why-does-0-1-zero-factorial-explained-simply
Why Does 0! = 1? Zero Factorial Explained Simply
November 18, 2025 - Find expert-backed shopping guides and top product picks from DHgate. Make smarter decisions with curated insights tailored for international buyers.
TikTok
tiktok.com › discover › why-is-0-factorial-1
Why Is 0 Factorial 1 | TikTok
3 weeks ago - Why is 0 factorial equal to 1 and not 0? That's a question my students often ask me when they're first learning about factorials. One explanation is that, by convention, empty products in math are defined to be 1, so that math doesn't break down.
Medium
thewalkingtemple.medium.com › why-exactly-is-zero-factorial-equal-to-one-570a31ea2949
Why Exactly Is Zero Factorial Equal To One? | by Hemanth | Medium
May 25, 2022 - It is common knowledge that zero factorial is equal to one. Everyone learns this in math class in middle school. However, when we delve into it further, most of us might realise that our understanding of this concept is rather shaky. The general challenge we have with mathematics is that we take things for granted. We often don’t wish to come across as stupid for asking ‘dumb’-sounding questions. If the teacher says 0! = 1...