If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Zero factorial or 0!

Learn all about factorials! Factorials are a quick way to represent multiplying a number by all the smaller positive integers down to one. This video also shows why mathematicians have defined zero factorial as one, instead of zero, to make the formula for permutations work in all cases.

Want to join the conversation?

  • duskpin ultimate style avatar for user Abhigyan Basak
    Is there any proof that 0 factorial equals 1?
    (41 votes)
    Default Khan Academy avatar avatar for user
  • male robot donald style avatar for user Alaska Mani
    Is it possible to have 2 factorial signs -- like 5!! ? Would that be 5! x 5! ?
    (14 votes)
    Default Khan Academy avatar avatar for user
    • leaf blue style avatar for user Dr C
      That would be something called a "multifactorial", which, for the case of two factorials is called a double factorial. For a number n, this is defined as:
      n!! = n * (n-2) * (n-4) * .... 

      Basically, instead of subtracting 1 each time, we subtract 2 (or if we had n!!! we'd subtract 3, and so forth). If we have k factorial signs (and hence subtract k each time), the sequence of multiplication ends when we can't subtract k without getting a negative number.
      (37 votes)
  • duskpin tree style avatar for user Francis Albert Calas
    Can you give an example where 0! can actually be used? Like for example the chair arrangement? Thanks
    (4 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user keanebrody8585
    I think its also worth noting there is only one permutation of a set of zero items.

    Here's an example:
    (14 votes)
    Default Khan Academy avatar avatar for user
  • winston default style avatar for user Soham Mukherjee
    Guys I have an alternate way....
    well we know,
    n! = n(n-1)! as per defination of factorial.
    so if we substitute n=1,we get,
    => 1!=1x(0!)
    hence, 0!=1 [as 1!=1]

    Please tell me if this method could also be acceptable or not and do share your views...I was playing with formulae and accidentally came up with this...
    have a good day
    (8 votes)
    Default Khan Academy avatar avatar for user
  • stelly orange style avatar for user Krishna Phalgun
    Can we define n! = (n+1)!/(n+1), if we substitute 0 in the above formula we can get 0! = 1.
    (5 votes)
    Default Khan Academy avatar avatar for user
  • piceratops ultimate style avatar for user Anirudh
    if 0! is 1 and 1! is 1 by obvious logic 1 should be equal to 0. What are the terms and conditions for such a scenario when x! is equal to y!
    (0 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user abassan
      If x!=y!, it doesn't follow that x-y. I know that because otherwise 0 would equal 1 and it doesn't.
      I would conclude that if x!=y! it follows that x=y unless x or y is 0 or 1.
      I have never seen an equation where x! = y! so it doesn't come up much.
      (2 votes)
  • aqualine ultimate style avatar for user Aryan Metha
    Is this a correct reasoning:
    Suppose you have n people and 0 chairs, there is only one combination possible that no one sits. so 0!=1
    (5 votes)
    Default Khan Academy avatar avatar for user
  • spunky sam blue style avatar for user Revan
    Cant we do this concepts with rational number ? I ma thinking why cant N go down below zero ?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • leaf blue style avatar for user Matthew Daly
      The only formulas you have at your disposal at the moment is (n+1)! = (n+1) n! and 1! = 1. Using this with n=0, we would get 1! = (1)(0!) or 0! = 1!/1, so there's nothing too unnatural about declaring from that that 0! = 1 (and the more time you spend learning math, the more it will seem to be the correct choice intuitively). Now let's try the same trick to define (-1)!. Setting n = -1 in our formula above, we get 0! = (0) (-1)! or (-1)! = 0!/0. But now we're in undefined land, because you can't divide by zero, so the factorial function cannot be extended to negative integers.

      Can you extend the factorial function to rational numbers (aside from the negative integers)? In theory, yes, but we don't have the tools in precalculus to talk about them. For instance, we could say that 1.5! = (1.5) 0.5!, but since we don't have any good ideas about what either of those factorials should be, we can't define any of them. When you get through calculus, you'll be able to understand a very awesome and weird function called the gamma function that actually accomplishes this task, but that's a conversation for another day. ^_^

      (5 votes)
  • male robot hal style avatar for user K.A.R.R3000.Shuvodeep
    Well, I was thinking a bit.......may be I am wrong.....
    4!=4x3x2x1 = 5!/5
    3!=3x2x1 = 4!/4
    2!=2x1 = 3!/3
    1!=1 = 2!/1
    As we are decreasing the number on the LHS(i.e before '!' symbol) by 1, we are dividing the RHS by that number......4! decreased to 3!, we are just dividing it by the number 4 on the RHS......
    Continuing this sequence 0! = 1!/1 = 1........
    (2 votes)
    Default Khan Academy avatar avatar for user

Video transcript

- If you've been paying particularly close attention to our use of the factorial operator in the videos on permutations and combinations, you may or may not have noticed something that might be interesting. So let's just review factorial a little bit. So if I were to say n factorial, that of course is going to be n times n minus one times n minus two, and I would just keep going down until I go to times one. So I would keep decrementing n until I get to one, and then I would multiply all those things together. So, for example, and all of this is review. If I were to say three factorial, that's going to be three times two times one. If I were to say two factorial, that's going to be two times one. One factorial, by that logic, I just keep decrementing until I get to one, but I don't even have to decrement here, I'm already at one. So I just multiply one. Now what about zero factorial, this is interesting. Zero factorial. So one logical thing is to say, maybe zero factorial is zero. I'm just starting with itself, it's already below one. Maybe it is zero. Now what we will see is that this is not the case that mathematics, mathematicians have decided. This is what's interesting, the factorial operation, this is something that humans have invented, that they think is just an interesting thing, it's a useful notation. So they can define what it does. And mathematicians have found it far more useful to define zero factorial as something else. To define zero factorial as... And there's a little bit of a drumroll here. They believe zero factorial should be one. And I know, based on the reasoning, the conceptual reasoning of this, this doesn't make any sense. But since we've already been exposed a little bit to permutations, I'll show you why this is a useful concept, especially in the world of permutations and combinations. Which is, frankly, where factorial shows up the most. Most of the cases that I've ever seen factorial in anything has been in the situations of permutations and combinations. And in a few other things, but mainly permutations and combinations. So let's review a little bit. We've said that if we have n things and we want to figure out the number of ways to permute them into k spaces, it's going to be n factorial over n minus k factorial. Now we've also said that if we had n things that we want to permute into n places, well this really should just be n factorial. Let's just do this. This is the first place, this is the second place, this is the third place. All the way, you get to the nth place. There would be n possibilities for who's in the first position, or which object is in the first position. And then for each of those possibilities, there would be n minus one possibilities for which object you choose to put in the second position, because you've already put one into that position. Now for each of these n times n minus one possibilities, where you've placed two things, there would be n minus two possibilities of what goes in the third position, and then you would just go all the way down to one. And this thing right over here is exactly what we wrote over here. This is equal to n factorial. But if we directly applied this formula, this would need to be n factorial over n minus n factorial. Then you might see why this is interesting. Because this is going to be n factorial over zero factorial. So in order for this formula to apply, even in the case where k is equal to n, which is this one right over here, and for that to be consistent with just plain old logic, zero factorial needs to be equal to one. And so the mathematics community has decided, this thing we've constructed called factorial, we said you put an exclamation mark behind something, in all of our heads, we say you kind of count down that number all the way to one and you keep multiplying them for zero, we're just going to define this. We're just going to define, make a mathematical definition. We're just going to say zero factorial is equal to one. And it's actually quite useful.