Combinations and its Formula Video Lecture | Mathematics for ACT

- The first time you're
exposed to permutations
and combinations it takes a little bit
to get your brain around it,
so I think it never hurts
to do as many examples.
But each incremental
example I'm gonna review
what we've done before, but hopefully go
a little bit further.
So let's just take another example.
This is in the same vein.
In videos after this I'll
start using other examples,
other than just people sitting in chairs,
but let's just stick with it for now.
Let's say we have six people again.
Person A, B, C,
D, E, and F,
so we have six people.
Now let's put them into four chairs.
We can go through this fairly quickly,
one, two, three, four chairs.
We've seen this show multiple times.
How many ways, how many
permutations are there
of putting these six
people into four chairs?
Well the first chair, if
we seat them in order,
we might as well, we
can say well there'd be
six possibilities here.
For each of those six possibilities
there would be five possibilities
of who we'd put here,
because one person's already sitting down.
For each of these 30 possibilities
of seating these first two people,
there'd be four possibilities
of who we put in chair number three.
Then for each of these, what
is this, 120 possibilities
there would be three possibilities
of who we put in chair four.
This six times five times four times three
is the number of permutations.
We've seen in one of the
early videos on permutations
that, when we talk about
the permutation formula,
one way to write this, if we
wanted to write in terms of
factorial, we could write this as
six factorial, six factorial
which is going to be equal to
six times five times four times
three times two times one,
but we wanna get rid of the two times one.
We're gonna divide that,
we're gonna divide that.
Now what's two times one?
Two times one is two factorial.
Where do we get that?
We wanted the first four,
the first four factors of six factorial.
That's where the four came from.
We wanted the first four
factors, so the way we got two
is we said six minus four.
Six minus four,
that's gonna give us the number
that we wanna get rid of.
We wanted to get rid of two
or the factors we wanna get rid of.
That's going to give us two factorial.
If we do six minus four factorial.
That's going to give us two factorial,
which is two times one.
Then these cancel out, and we are all set.
This is one way, I put in
the particular numbers here,
but this is a review of
the permutations formula,
where people say, "Hey, if I'm saying n,
"if I'm taking n things,
then I want to figure out
"how many permutations are there
"of putting them into let's say k spots,
"it's going to be equal to n factorial
"over n minus k factorial."
That's exactly what we did over here,
where six is n
and k, or four is k.
Four is k.
Actually let me color code the whole thing
so that we see the parallel.
All of that is review.
Then we went into the
world of combinations.
In the world of combinations,
we said permutations make a difference
between who's sitting in what chair.
For example, in the permutations world,
and this is all review, we've covered this
in the combinations video,
in the permutations world
A, B, C, D,
and D, A, B, C,
these would be two different permutations
that's being counted in
whatever number this is.
This is what?
This is 30 times twelve.
This is equal to 360.
Each of these, this is one permutation,
this is another permutation,
and if we keep doing it
we would count up to 360.
But we learned in combinations,
when we're thinking about combinations,
let me write combinations.
If we're saying n choose,
n choose k,
or how many combinations are there?
If we take k things, and
we just wanna figure out
how many combinatio-
If we start with n, if
we have pool of n things,
and we wanna say how many combinations
of k things are there,
then we would count these
as the same combination.
What we really wanna do is we wanna take
the number of permutations there are.
We wanna take the number
of permutations there are,
which is equal to n factorial over
n minus k factorial,
over n minus k factorial,
and we wanna divide by the number of ways
that you can arrange four people.
Once again, this takes-
I remember the first time I learned
it took my brain a little while.
If it's taking you a little
while to think about it,
not a big deal.
It can be confusing at first,
but hopefully if you
keep thinking about it
hopefully you will see
clarity at some moment.
What we wanna do is we wanna
divide by all of the ways
that you can arrange four things.
Cuz once again, in the permutations
it's counting all of the
different arrangements
of four things, but we don't wanna count
all of those different
arrangements of four things.
We wanna just say they're
all one combination.
We wanna divide by the number of ways
to arrange four things.
Or the number of ways to arrange k things.
Let me write this down.
What is the number of ways,
number of ways,
to arrange
k things,
k things,
in k spots.
I encourage you to pause the video,
because this actually a review
from the first permutation video.
If you have k spots, let me do it
so if this is the first
spot, the second spot,
third spot, and then
you're gonna go all the way
to the kth spot.
For the first spot
there could be k possibilities.
There's k things that
could take the first spot.
For each of those k possibilities,
how many things can be in the second spot?
It's gonna be k minus one,
because you already put,
you already put something
in the first spot.
Then over here, what's it gonna be?
K minus two
all the way to the last spot,
there's only one thing that
can be put in the last spot.
What is this thing here?
K times k minus one times k minus two
times k minus three all
the way down to one.
This is just equal to k factorial.
The number of ways to
arrange k things in k spots,
k factorial.
The number of ways to arrange
four things in four spots,
that's four factorial.
The number of ways to arrange three things
in three spots, it's three factorial.
We could just divide this.
We could just divide this by k factorial.
This would get us, this would get us,
n factorial
divided by k factorial,
k factorial
times,
times n minus k factorial,
n minus k,
n minus k,
I'll put the factorial right over there.
This right over here is the formula.
This right over here is the
formula for combinations.
Sometimes this is also called
the binomial coefficient.
People always call this n choose k.
They'll also write it like this,
n choose k, especially
when they're thinking
in terms of binomial coefficients.
I got into kind of an
abstract tangent here,
when I started getting
into the general formula.
Let's go back to our example.
In our example we saw
that there was 360 ways
of seating six people into four chairs.
What if we didn't care about
who's sitting in which chairs
and just wanna say,
"How many ways are there
"to choose four people
"from a pool of six?"
That would be
that would be how many ways are there.
That would be six.
How many combinations
if I'm starting with a pool of six,
how many combinations are there?
How many combinations are there
for selecting four?
Another way of thinking about it is
how many ways are there to,
from a pool of six items,
people in this example,
how many ways are there
to choose four of them.
That is going to be, we could do it-
I'll apply the formula first,
and then I'll reason through it.
And like I always say,
I'm not a huge fan of the formula.
Every time I revisit it after a few years,
actually just rethink about it,
as opposed to memorizing it,
because memorizing is a
good way to not understand
what's actually going on,
but if we just apply the formula here,
I really want you to understand
what's happening in the formula,
it would be six factorial
over four factorial,
over four factorial,
times six minus four factorial,
six, oops let me just,
This is six minus four factorial,
so this part right over here,
six minus four fa-
Let me write it out because I know
this can be a little bit confusing
the first time you see it.
So six minus four factorial,
factorial, which is equal to,
which is equal to six
factorial over four factorial,
over four factorial, times
this thing right over here
is two factorial, times two factorial,
which is going to be equal to, we can just
write out the factorials,
six times five times four
times three times two times one,
over four.
Four time three times two times one
times,
times two times one.
Of course that's going
to cancel with that.
Then the one doesn't
really change the value,
so let me get rid of this one here.
Let's see, this three can
cancel with this three.
This four could cancel with this four.
Then it's six divided by
two is going to be three.
So we are just left with three times five.
We are left with,
we are left with, there's
fifteen combinations.
There's 360 permutations for putting
six people into four chairs,
but there's only 15 combinations,
because we're no longer counting
all of the different arrangements
for the same four people
in the four chairs.
We're saying, "Hey if
it's the same four people,
"that is now one combination."
You can see how many ways are there
to arrange four people into four chairs?
That's the four factorial
part right over here.
The four factorial part right over here,
which is four times three
times two times one,
which is 24.
We essentially just took the 360
divided by 24 to get 15.
Once again, I don't think
I can stress this enough.
I wanna make it clear
where this is coming from.
This right over here,
let me circle,
this piece right over here is
the number of permutations.
This is really just so you can get to
six times five times four times three,
which is exactly what we did up here,
where we reasoned through it.
Then we just wanna divide
by the number of ways
you can arrange four items in four spaces.