Zero factorial or 0!

- 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.