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
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!
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!
combinatorics - Why is 0 factorial equal to 1? Is there any pure basic mathematical proof? - Mathematics Stack Exchange
If you are starting from the "usual" definition of the factorial, in my opinion it is best to take the statement $0! = 1$ as a part of the definition of the factorial function, as anything else would require proofs using the factorial to include special cases for $0!$ and $1!$. It's a definition ... More on math.stackexchange.com
Explain zero factorial.
Hint: In mathematics, zero factorial is the expression that means to arrange the data containing no values. The value of n! is given by $n!=n\times \left( n-1 \right)\times \left( n-2 \right)\times \ldots \ldots \ldots \times 1$ . The given equation can also be written as $n!=n\times \left( n-1 \right)!$ . we need to substitute n=1 to get the value of 0... More on vedantu.com
1
3
January 17, 2026how is 0 zero factorial equal to 1 mathematical proof 33268
VIDEO ANSWER: Hello, so understanding here why 0 factorial is equal to 1. Okay, so this can be understood from the definition of the factorial function. And so factorials are going to be defined primarily for non -negative integers. So for any More on numerade.com
1
ELI5: Why does 0! (zero factorial) equal 1?
English: In how many ways can you arrange no objects? Obviously one way.
Mathmatish:
It's a continuision of the definition.
If we accept the definition that
x! = x * x-1 * x-2 * x-3 ... * 2 * 1
then that would also infer the definition
x! = (x-1)! * x
which can be rearranged as
(x-1)! = x! / x
So, now we have a rule. Now, let's start counting down from 3!
3! = 4!/4 = 24/4 = 6 2! = 3!/3 = 6/3 = 2 1! = 2!/2 = 2/2 = 1 0! = 1!/1 = 1/1 = 1
We've thus shown that 0! = 1 and that this definition is logical and fits perfectly into the series.
More on reddit.com 8
2
September 30, 2014Videos
01:01
Proof that zero factorial equals one (0!=1) - YouTube
03:30
0!=1 PROOF | Zero Factorial is Equal to One - YouTube
Zero factorial or 0! (video) | Probability
02:05
0!=1 Proof | How to Prove Zero Factorial is Equal to One | Zero ...
04:16
Intuiting Why Zero Factorial Is One // Math Mini - YouTube
GeeksforGeeks
geeksforgeeks.org › mathematics › zero-factorial
What is Zero Factorial (0!): Meaning, Calculation, Applications - GeeksforGeeks
December 7, 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 ...
ThoughtCo
thoughtco.com › why-does-zero-factorial-equal-one-3126598
Why Does Zero Factorial Equal One?
May 18, 2024 - To find the value of zero factorial, we ask, “How many ways can we order a set with no elements?” Here we need to stretch our thinking a little bit. Even though there is nothing to put in an order, there is one way to do this.
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
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.
Wikipedia
en.wikipedia.org › wiki › Factorial
Factorial - Wikipedia
1 week ago - {\displaystyle 5!=5\times 4!=5\times 4\times 3\times 2\times 1=120.} The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical ...
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.
. The given equation can also be written as 
. we need to substitute n=1 to get the value of 0!Complete step by step answer:Factorial of a number in mathematics is the product of all the positive numbers less than or equal to a number. The multiplication happens to a given number down to the number one or till the number one is reached.Example: Factorial of n is n! and the value of n! is 
Definition 1:In mathematics, zero factorial is the expression that means to arrange the data containing no values. Factorial is used to define possible data sets in a sequence also known as permutation. Order is important in the case of permutations. As per the same, if there are no values like in an empty or zero set there is still a single arrangement possible.As there is no data to arrange, the value becomes eventually equal to one.Definition 2:Combinations usually are the number of ways the objects can be selected without replacement.Order is not usually a constraint in combinations, unlike permutations.Factorial of a number in mathematics is the product of all the positive numbers less than or equal to a number. But there are no positive values less than zero so the data set cannot be arranged which counts as the possible combination of how data can be arranged (it cannot).Thus, 0! = 1.Definition 3:Factorial of a number in mathematics is the product of all the positive numbers less than or equal to a number. Example: Factorial of n is n! and the value of n! is 
The value of n! from the above can be also written as 
Considering the value of n equal to 1,
The value of LHS should be equal to RHS as 1! is always equal to 1!For the above condition to be true, The value of 0! must be equal to 1.The value of 0! =1.Note: The factorial of a number is denoted by an exclamation mark. Factorial of a number only deals with natural numbers so zero is omitted. The multiplication of any factorial takes place down to 1 and not zero. Factorials are usually used in the context of solving permutations and combinations. The Math Doctors
themathdoctors.org › zero-factorial-why-does-0-1
Zero Factorial: Why Does 0! = 1 ? – The Math Doctors
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?? Furthermore, the first use of the factorial is to answer permutation problems: \(n!\) is the same as \(_n\text{P}_n\), the number of ways to arrange all of n objects a row.
YouTube
youtube.com › watch
0!=1 PROOF | Zero Factorial is Equal to One
Learn the proof to why 0 factorial is equal to 1. Quick and simple explanation by PreMath.com
Published May 24, 2021
YouTube
m.youtube.com › watch
Prove that Zero Factorial is Equal to One
Share your videos with friends, family, and the world
Numerade
numerade.com › ask our educators › algebra › how is 0 zero factorial equal to 1 mathematical proof 33268
how is 0 zero factorial equal to 1 mathematical proof 33268
VIDEO ANSWER: Hello, so understanding here why 0 factorial is equal to 1. Okay, so this can be understood from the definition of the factorial function. And so factorials are going to be defined primarily for non -negative integers. So for any
Published October 6, 2023 Views 0
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 - Why Exactly Is Zero Factorial Equal To One? Once you get this, you won’t need to take it for granted anymore! It is common knowledge that zero factorial is equal to one. Everyone learns this in …
Math Answers
math.answers.com › algebra › What_is_zero_factorial
What is zero factorial? - Answers
Zero factorial, written as 0!, equals 1.
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.
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.
Quora
quora.com › What-will-be-the-value-of-0-factorial-with-proof
What is the value of zero factorial (0!)?
Answer (1 of 55): 0!=1 Now, you could be asking yourself “Wait a minute, isn't the factorial of a number supposed to be the multiplication of every positive integral from that number down to 1? Clearly 0! Is 0" Well, in theory yes, but 0 is lower that 1, so you can not go from 0 DOWN to 1, righ...
TikTok
tiktok.com › discover › proof-of-zero-factorial
Proof of Zero Factorial
November 17, 2025 - Family Spring Break - TikTok GO campaign. Campaign dates are Mar 5th - 18th. Post about things to do! Up to $12,000 in rewards for creators · Spring is here! Join TikTok Shop’s Spring Statements campaign! Post short videos of your favorite spring fashion pieces and trendiest spring looks.
YouTube
youtube.com › watch
Prove that Zero Factorial is Equal to One - YouTube
Counting Principles Playlist: https://www.youtube.com/watch?v=-zjqpzBzV54&list=PLJ-ma5dJyAqqvQaevRUhshT_EmYvF52oq&index=3Best Videos on Combination: https://
Published September 9, 2016
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 - Only from the definition of a factorial number and by dividing both sides by n one has: n!/n (n-1)! or (n-1)! = n!/n when n=2 one has (2-1)! = 2!/2 or 1! = 2x1/2 or 1! = 1 when n=1 one has (1-1)! = 1!/1 or 0! = 1/1 or 0! = 1. = This is a proof ...