Imagine a shelf. On that shelf, put 3 cards: one that says 1. One that says 2. And one that says 3. Put them on the shelf in some order (e.g. 2,1,3). How many orderings are possible? 3! = 6, right? Exactly. That’s how many different ways my shelf could look different. Now imagine we had 1 card instead of 3. How many ways can the shelf look? It’ll look like this: ____ [ 1 ] ____ with just the one card with a 1 on it. That’s the only possibility for the shelf. 1 factorial is 1. Now imagine we, in fact, have 0 cards. What’s 0 factorial? How many different ways can the shelf look? Only one: ________. There is one way to order zero objects. The way that produces an empty shelf. Answer from R0KK3R on reddit.com
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 - It might sound strange, but this “empty arrangement” is still considered a valid arrangement. In combinatorics, an empty set has exactly one subset: itself, and there’s exactly one way to arrange nothing. So, we define 0! = 1 to reflect that.
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 - Factorial is not defined by how many ways there to order n objects. Ohh, just watch me. Definition: For each nonnegative integer n\in N let N_n denote the set {1,2,...,n} and let S_n be the permutation group of N_n. We define the factorial n! of n to be the order |S_n| of the group S_n. ... From that, N_0 = ∅ and we have that S_0 is the permutation group of ∅, and since there does exist a bijection from ∅ to ∅ thus S_0 is inhabited.
Videos
07:34
Why 0!=1? How about (-1)! = ? Reddit r/learnmath, zero factorial ...
06:05
Why is 0! = 1? - YouTube
02:17
Why is 0! equal to 1? - YouTube
03:30
0!=1 PROOF | Zero Factorial is Equal to One - YouTube
04:16
Intuiting Why Zero Factorial Is One // Math Mini - YouTube
Reddit
reddit.com › r/explainlikeimfive › eli5: why is 0!=1?
r/explainlikeimfive on Reddit: ELI5: why is 0!=1?
July 27, 2023 - If you define n! to be the product ... set. 1 makes the most sense for the product of the empty set because 1 multiplied by any number is that other number (it's the multiplicative identity)....
ThoughtCo
thoughtco.com › why-does-zero-factorial-equal-one-3126598
Why Does Zero Factorial Equal One?
May 18, 2024 - A set with two elements has two permutations: {a, b} can be arranged as a, b or b, a. This corresponds to 2! = 2. A set with one element has a single permutation, as the element 1 in the set {1} can only be ordered in one way.
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??
Top answer 1 of 5
16
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.
2 of 5
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
Free Math Help Forum
freemathhelp.com › forums › free math help › arithmetic
Why is 0!=1? | Free Math Help Forum
January 1, 2022 - It is defined as: "the product of all positive integers from 1 to n". I don't think it tells us anything about n<1, i.e. 0 or negatives. With these three arguments so far, I see that it's convenient to have 0!=1, but not necessary hence why question can I define my own 0!
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 ...
Facebook
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Why is 0! equal to 1?
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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.
YouTube
youtube.com › cowan academy
Why is 0! = 1 (Proof) - YouTube
Why is 0 factorial equal to 1 Proof Beginning with the definition of factorials we can work our way to a proof where 0! = 1 is mathematically proven. In the ...
Published March 5, 2018 Views 101K
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....
TikTok
tiktok.com › mr trivedi does maths (@mrtrivedidoesmaths) | tiktok › why is 0! equal to 1? let me explain #factorials #factorial #maths
Why is 0! Equal to 1? Let me explain #factorials #factorial #maths | TikTok
17.2K Likes, 184 Comments. TikTok video from Mr Trivedi Does Maths (@mrtrivedidoesmaths): “Why is 0! Equal to 1? Let me explain #factorials #factorial #maths”. 0! = 10!=1???original sound - Mr Trivedi Does Maths.
Wikipedia
en.wikipedia.org › wiki › Natural_number
Natural number - Wikipedia
1 day ago - The axiom of induction: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number. These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0.
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!
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
April 28, 2022 - What is the rationale for defining 0 factorial to be 1?AnswerThe defining 0 factorial to be 1 is not a rationale."Why is zero factorial equal to one?" is a problem that one has to prove.When 0 factorial to be 1 to be proved,the defining 0 factorial ...
Wikipedia
en.wikipedia.org › wiki › Euler's_identity
Euler's identity - Wikipedia
17 hours ago - Euler's identity is the case where n = 2. A similar identity also applies to quaternion exponential: let {i, j, k} be the basis quaternions; then, ... The same formula applies to octonions, with a zero real part and a norm equal to 1. These formulas are a direct generalization of Euler's identity, since ... {\displaystyle -i} are the only complex numbers with a zero real part and a norm (absolute value) equal to 1.
Quora
quora.com › Why-do-we-take-0-as-1-Is-it-an-inconsistency-in-mathematics
Why do we take 0! as 1 ? Is it an inconsistency in mathematics? - Quora
Answer (1 of 12): It’s not an inconsistency. blackpenredpen explained this in detail in “Is this true: (0!+0!)^(0!+0!)x(0!+0!)^(0!+0!)=16? Reddit math problem”. Let’s say we start with 5!. That is 5x4x3x2x1, which is 120. Next we go to 4!. That is 4x3x2x1, which is 24.
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.
GeeksforGeeks
geeksforgeeks.org › dsa › why-factorial-of-0-is-1
GFact | Why Factorial of 0 is 1? - GeeksforGeeks
October 4, 2023 - Example: Factorial of n is n! and the value of n! is n! = n × (n−1) × (n−2) ×………× 1. In mathematics, zero factorial(0!) is the expression that means to arrange the data containing no values.