Let's say I have 3 chairs.
They'll just be blanks: 1, 2, and 3.
3 chairs, and I have 7 people.
I'll call them people A, B, C, D, E-- how many is that?
That's 5-- F, G.
So we have 7 people and what I want to know is how many
different ways can these 7 people sit in these 3 chairs?
Well, before anyone sits down if we just pick a chair, these
are just arbitrary labels for the chairs, but let's start at
the left with chair number one.
If no one is sitting down, how many different people could
sit down in chair one?
Well, 7 people are standing up, no one is sitting in any of the
chairs, so 7 different people could sit in chair one.
So there are 7 possibilities for chair one.
So 1 of those 7 possibilities is going to sit
down in chair one.
So how many people are left to sit in chair two?
Well, 1 less than 7, right?
6 people are going to be left to sit in chair two.
And then of course, if 2 people are already sitting down, how
many people are left to sit in chair three?
Well, there's going to be 5 possibilities.
So a way to think about it, for each of these possibilities,
there are 6 possibilities in this chair and 5 possibilities
in that chair for each of those 6.
So you would multiply all of the possibilities.
Hopefully that makes some sense.
We'll try it out with some examples.
So the total number of possibilities in this
case is 7 times 6 times 5, and what is that?
6 times 5 is 30.
30 times 7 equals 210.
There's 210 possibilities.
That's a lot of possibilities.
Let's do a slightly smaller example.
I mean we don't have to do people in chairs.
Let's say we have cups.
And I'll do the cups in a different color.
So this is 1 cup, cup one.
And then I have cup two.
And then I have 3 balls.
I have a magenta ball, I have a brown ball, and
I have a yellow ball.
And I want to know, how many different ways can I put these
3 balls into these 2 cups?
Well, assuming that I haven't put any of the balls into any
of the cups as yet, how many balls-- well, let's just
start with cup two.
I just want to show you that you don't have to start at the
cup that happens to be labeled number one sitting on the left.
If we haven't put any balls into any of the cups, how
many can we put in cup two?
Well, there's 3 possibilities, right?
And for each of those 3 then, how many can
we put into cup one?
Well, we would've put 1 into here, so there will be 2
left to put into cup two.
So it'd be 3 times 2 possibilities, which
is equal to 6.
And let's see if we can draw them out.
Let me number these cups because it's going to be
too difficult to keep switching colors.
So let's say this is A, B, and C.
I'm going to show you that there are 6 ways of putting
these 3 balls into these 2 cups.
So it could be A in cup one, B in cup two.
It could be B in cup one, A in cup two.
Right, you could just switch them.
So we actually care which ball goes into what cup.
Not just is a ball in a cup.
It could be A with C.
A in cup one, C in cup two.
It could be C in cup one, A in cup two.
And then finally, you could just have B and C.
B in cup one, C in cup two.
Or C in cup one, B in cup two.
And notice, if we just cared which of the balls were picked,
but we didn't care whether they were in cup one or cup two, we
could get rid of this bottom line and there would be 1/2
as many ways to place them.
But when we care where the balls are-- we care whether
they're in cup one or cup two, or in this case we care if
they're in seat one, two, or three.
The ways in which we're ordering, we call
these permutations.
And up here, the way we would write this permutation, we
would say, well, how many ways can we fit 7 people into 3
chairs if we care about what chair they're sitting in?
So we could write that as 7 people-- and the p isn't
for people, it's for permutations-- into 3 chairs.
And another notation for it, you write a big P and you
say, how many ways can I put 7 things into 3 spaces?
Or it could be written as, how many ways could you put
7 things into 3 spaces?
And in this case, we got the answer is 210.
And actually, it will always be 210.
And I'll show you why in a second.
How would we write this?
Well, this would be-- I'll do it in green.
We had 3 things and we put it into 2 spaces.
So 3 things into 2 spaces.
That's the same thing as 3 things into 2 spaces, which
is equal to 3 things into 2 spaces.
I'm just showing you the different notations, and we
figured out that that was 6.
Let's see if we can figure out a general formula if
we wanted to evaluate nPk.
How many ways can we fit n things into k spaces?
So if we think about it, let's just use some analogy.
We'll have k spots, so it'll be 1 spot, 2 spots, 3 spots,
all the way to k spots.
In the first spot there's n possibilities, just like we
did in the last two examples.
There's n possibilities there.
Then, since 1 person or 1 thing will be placed in spot one,
there's going to be n minus 1 possibilities for spot two.
And similarly, n minus 2 possibilities for spot three.
And then, we could follow that pattern all the way, and how
many possibilities are there going to be for spot k?
Well in everything it's n minus 1 thing less than the
spot number in our case.
I mean I guess we could have started with spot 0, but this
is better because we actually numbered all the way to k.
So it will be n minus k minus 1.
And that might seem complicated, but
it makes sense.
When we did 7 things into 3 spots we went 7 times 6 times--
we literally put in the top three of you could almost
say from the factorial.
7 factorial will be 7 times 6 times 5.
I think this will help.
If this was just 3 was so 7 times 6 times 5 times 4
times 3 times 2 times 1.
That's 7 factorial.
But we only took the first 3 of the factorial.
And similarly, 6-- well, 6 is also 3 factorial.
But another way you could view it is we only took the first 2
of 3 factorial So what we're saying here is we just take
the first k of n factorial.
So is there any easy way to write that?
To write the first k of n factorial.
Why sure.
We could write it as n factorial and the formula's
almost harder than what it's saying.
We could write it as n factorial-- it;s a lower
case n-- n factorial over n minus k factorial.
Let's see if that works.
Well, what's going to happen?
So in this case, n factorial was 7.
So it was 7 times 6 times 5 times 4 times
3 times 2 times 1.
And then what was n minus k factorial?
Well, in our example, k was the number of spots,
and there were 3 spots.
So 7 minus 3 is 4, so 4 factorial.
4 times 3 times 2 times 1.
And of course, this will cancel.
Let me do that in a different color.
This will cancel with this, and so you're just left with
the first 3 of 7 factorial.
The 3 largest components of 7 factorial.
I always rederive this because all I think of is what
we did originally.
I was like, oh, OK.
If we're doing permutations, I just start the factorial
until I run out of spots.
I do 7 times 6 times 5.
And let's see if it works out here too.
So what is 3 factorial because we have 3 different things?
3 factorial over-- and then 3 minus 2 factorial.
Well, that's just 1 factorial.
So it becomes 3 times 2 times 1 over 1.
And of course, the 1's cancel out and we got the 3 times 2.
Hopefully that make sense.
A lot of people, they learn the permutation notation and they
say, oh, that's n factorial over n minus k factorial, where
n is the number of things I need to place and n minus
the whole thing factorial.
And k is the number of spots I have.
And they just memorize it.
But all this is saying is you start doing the n factorial,
but you only do the first 3 components of the factorial.
And it makes sense just from how we started off the video.
If you 7 things, well, 7 things could go here, 6 things could
go here, 5 things could go here.
No more spots?
Just multiply them.
And that's all this formula is saying.
Hopefully I didn't confuse you, and I will see you in the
next video on combinations.