Factorial and counting seat arrangements

- So, let's say that we have a round table

and we have three chairs
around that round table.

This is one chair right over here,

this is another chair,

and that is another chair.

We can number the chairs;

that is chair one,

that is chair two,

and that is chair three.

Now let's assume that
there are three people

who want to sit in these three chairs,

so there is Person A,

there is Person B,

and there is Person C.

What we want to do is we want to count

the number of ways that these three people

can sit in these three chairs.

What do I mean by that?

Well, if A sits in chair
one, B sits in chair three,

and C sits in chair two, that is one

scenario right over there.

So this is one of the possible scenarios,

A in seat one, C in seat
two, B in seat three.

Another scenario could be B in seat one,

maybe C still sits in seat
two, C still sits there,

and now A would be in seat three.

So this is one scenario and two scenarios.

My question to you is how many

scenarios are there like this?

I encourage you to pause the video

and try to think through it.

I'm assuming you've had a go at it,

now let's work through it together.

I want to do it very systematically

so that we don't forget
any of the scenarios.

The way I'm going to
do it is, I'm going to

create three blanks that
represent each of the chairs.

So here, it's in a circle,
but we could just say

that blank is chair one,
that's the blank for chair two,

and that is the blank for chair three.

I'm going to start with seat
one, and I'm going to say,

"Well, what are the different scenarios?"

A could sit there, B could
sit there, C could sit there.

Then for each of those, figure
out who could sit in seat two

and then for each of those,

figure out who could sit in seat three.

So let's do that.

First, what are the scenarios
of who could sit in seat one?

Maybe I'll write it this way,

maybe I'll do seat one.

So right now, we're only
going to fill seat one.

Well A could sit in seat one,

in which case we haven't filled
seat two or seat three yet.

So that's two or three.

B could sit in seat one,
and we haven't figured out

who sits in seat two or seat three yet.

Then we could have C sitting in seat one,

and we still have to figure
out who's going to sit

in seat two and seat three.

For each of these, let's figure out who

could be sitting in seat two.

So, seats one and two.

For this one right over here,

we have A in seat one for sure.

We could either have
B sitting in seat two,

in which case, I'll leave this
blank although you can use

a little bit of deductive
reasoning to figure out

who's going to be sitting in seat three.

So I'll leave seat three blank.

Or, you could have C sitting in seat two.

We still haven't figured out
who's sitting in seat three,

although once again, a little bit

of deductive reasoning might tell us.

So these are the scenarios
where A is in seat one.

Now what about when B is in seat one?

We could put A in seat two, and we still

need to figure out who's in seat three.

Or, we could put C in seat two,

and we still need to figure out

who's going to be in seat three.

Finally, let's look at C,
where C is in seat one.

You could either put A in seat two or

you could put B in seat two.

Now let's fill out all three of the seats.

So seats one, two,

and three.

For this scenario right over here,

there's only one option that
you could put in seat three,

the only person who hasn't
sat down yet is person C.

So this will be A, B, C.

And what would this be?

The only person who hasn't sat down

in this scenario yet to fill this chair,

the only option for this chair,

there's only one option here,

it's going to be person B.

It's A, C, B.

This one is going to be B, A, C.

C's the only person who could sit there,

A is the only person who could sit there.

B, C, A.

And then here, B is the only person

who could sit in seat three.

And here A is the only person

who could sit in seat three.

How many scenarios do we have?

We have one, two, three, four,

five, and six.

And six is our answer.

That's the number of
scenarios that you could have

the different people sitting
in the different chairs.

Now, I know what you're thinking,

"Okay Sal, this was a
scenario with three people

"and three chairs and I
might've been able just do this

"even if I didn't do it systematically.

"But what if I have many many more chairs?

"What if I had 60 chairs?"

Well, the number of scenarios
would get fairly large.

What if I had five people and five chairs?

Even this method right over here

would take up a lot of paper space,

or a lot of screen space.

What do we do in that scenario?

And the realization here is just

thinking about what happened here.

How many people could sit in seat one?

If we're going seat by seat,

how many possibilities were there

to put in seat one?

If you're seating in order,

if you haven't filled
any of the seats yet,

right at the beginning,
there's three possible people

who could sit in seat one.

You see it right over here, one

two, three, because there's
three possible people.

For each of those, how many
people could sit in seat two?

For each of them, let's
say the scenario where A

is sitting in seat one,
there's two possibilities

for who could sit in seat two.

Let me do this in a different color.

So this, right over here.

For each of these three,
there's two possibilities

of who could sit in seat two.

If A is sitting in seat
one, you could have

either B or C in seat two.

If B is sitting in seat one, you have

either A or C for seat two.

So for each of these three,
you have two possibilities

of who could sit in seat two.

So you have six possibilities
where you're filling in

three times two, or you're filling in

the scenarios for seats one and two.

And then for each of those,

how many scenarios are
who can sit in seat three?

For each of those, there was only

one possibility for who could sit

in seat three.

We see it right there, that's because

there's only one person left
who hasn't sat down yet.

So how many total possibilities are there?

Well, three times two
times one is equal to six.

And so if we were to do that
same exact thought exercise,

with say, five seats, let's
go through that exercise.

It's interesting.

So if we have one, two, three, four, five.

So this is five seats with
five people sitting down,

and we want to figure out
how many scenarios are there

for all the different people
and all the different seats.

For the first seat, there's five

different people who could sit in it.

For each of those scenarios,
there's four people

who haven't sat down yet who
could sit in the second seat.

For each of those scenarios,
there's going to be

three people who haven't sat down yet,

so those are the people who
could sit in the third seat

if we're filling out
the seats in this order.

For each of all of these scenarios,

there's two people left
who haven't sat down yet,

who could sit in the fourth seat.

And then for all of these scenarios,

there's only one person
left who hasn't sat down

who would have to sit in the fifth seat.

If you had this exact
same thing, but instead of

three chairs you had five chairs,

the number of scenarios would be

five times four times
three times two times one.

Which is what, 20 times six,

which is equal to 120 scenarios.

And so you see that the
scenarios grow fairly quickly.

And if you're wondering,
"Hey, this is kind of a neat

mathematical thing. Three
times two times one."

Or five times four times
three times two times one.

If you start with a number
and you multiply that number

times number one less than that

all the way down to getting to one,

that seems like neat kind of
fun mathematical operation.

And lucky for you, or
maybe unlucky for you,

because this might have
been your chance at fame,

this operation has already been defined.

It's called the factorial operation.

So this thing right over here,

this is the same thing as three factorial.

You write this little exclamation mark.

This right over here,
this is the same thing

as five factorial.

In general, if I said six factorial,

that would be equal to six times five

times four times three
times two times one.

Hopefully you enjoyed that.