- 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.