Binomial Theorem (part 3)

Intuition behind why binomial expansion involves combinatorics

In this video, I'm going to
attempt to give you an

intuition behind why
multiplying binomials involve

combinatorics

Why we actually have the
binomial coefficients

in there at all.

And I'm going to do
multiple colors.

The colors will actually be
non-arbitrary this time.

Just to give you an intuition.

So let's multiply a plus
b to the third power.

Well, a plus b to the third
power, that's a plus b times--

and I'm going to keep
switching colors.

You're going to have to bear
with me, but it should

hopefully be fruitful.-- Times
a plus b, times-- let me pick

an appropriately different
color, maybe a blue--

times a plus b.

And let's do this as a
distributive property.

This equals a times-- go
back in green-- a plus b.

With a different green.

Want to make sure I use the
right green, just because the

colors matter this time.

a plus b-- this is tedious,
but it's worth it--

plus b times a plus b.

And then all of that
times a plus b again.

And let's multiply
this inside part.

I'm just going to keep
it and multiply it out.

So it's going to be a times
green a-- we know they're all

the same a-- plus-- I should do
the pluses in a neutral color,

but it's OK-- plus a times--
you might be finding this

tedious, but it's going to pay
off in the end-- a times b.

Then we have plus b times a,
plus yellow b times green b.

And then all of that--
we're almost there.

We're almost there.

All of that times a plus b.

And it's essentially,
we're going to multiply

a times everything.

This blue a times all of
this, and plus this blue

b times all of this.

So let's multiply the blue
a times all of this.

So the first term will
be yellow a, green

a, and then blue a.

So it'll be a, a, a.

Oh it's a different blue, but
I think you get the point.

Plus this times blue a.

So yellow a, green b, and then
blue a, plus yellow b, green

a, blue a-- hopefully I'm
not confusing you-- baa.

b, a, a.

We're almost there.

Plus yellow b times
green b times blue a.

So we did all the
blue a's, finally.

There's a blue a.

Plus--.

Now we're going to do the
blue b times everything.

So it's yellow a times green a.

Right?

Yellow a, green a, then blue b,
times green a, times blue b--

almost there, I know this is
tedious-- times blue

b, plus yellow a.

Intuition doesn't
come easy, though.

Yellow a-- so we're on
this term-- yellow

a, green b, blue b.

So green b times blue b.

Now we're at yellow b-- the
good thing about the colors

is it's easy to keep
track of where we are.

Plus yellow b times
green a times blue b.

And then we're at the last one.

Plus yellow b, green
b, times blue b.

So this is the expansion
of a plus b to the

third power, right?

We haven't simplified
it at all, and I did

that for a reason.

Because you see that every term
here-- what's happening here?

Every term has exactly
one-- it's 3 numbers

being multiplied, right?

Every term is 3 numbers
being multiplied.

And it's one of, you know, the
yellow number comes from the

first-- from this
yellow a plus b.

The green number-- the
middle number-- comes from

this middle a plus b.

And then the blue number comes
from this right hand a plus b.

And you saw me, I went all
the way through it, right?

So hopefully you
believe this point.

So let's think about it a
couple of different ways.

To generate each of these terms
in the expansion of a plus b to

the third, we're picking either
a or b from-- from the yellow a

plus b, we're picking
either a or b.

Right?

We picked an a here,
we picked an a here.

We picked a b here, a b here,
an a here, an a here, a b--.

And the group from the green
a plus b, we're picking

either an a or a b.

And then from the blue a
plus b, we're picking

either an a or a b.

Right?

So essentially the expansion,
if you think about it, this

expansion-- we've essentially
done every way of choosing

3 different things.

Every way of picking either
an a or a b, from these

3 different terms.

And that ends up with
these-- what is this?

1, 2, 3, 4, 5, 6, 7, 8 terms.

Right?

Now let's make it a little
bit-- let's give you a

little bit more intuition
of what's going on.

And I think if you're starting
to realize why this is dealing

with permutations and
combinations, in particular.

Once we simplify
it, what happens?

This is a to the cubed, right?

That's the only a cubed term.

This is a squared b.

What are the other
a squared b terms?

Let's see, a squared b.

This is also a squared b.

So let me write down
all the a squared.

So let's see-- let's
see how many a squared

b terms there are.

I'll do it in a neutral color.

So this is a squared b.

This is ba squared, but
that's also a squared b.

What's another a squared b?

This is also a squared b,
right? a times a times b.

So there was 3 ways
to get a squared b.

And that's why when we
eventually write the expansion,

it's going to-- we know that
the coefficient in front of it

is 3a squared b, right?

The coefficient on the a
squared b term, when we

actually multiply it out.

And we've done that several
times already, when we did the

binomial theorem, which we
took to the third power.

So where did this 3 come from,
and why is that the same thing

as when we learned the
definition of the

binomial theorem?

Why is, you know, does it just
happen to be the case that

that's the same thing as 3
choose 2 times a squared b?

Well, no.

Think of it this way.

We already know that every
term here, every term in the

expansion, we're essentially
picking either an a or a b

from each of these, right?

We have to pick one term
from each of these.

So the way you could think of
it, for a squared b-- to get a

squared b-- we have to
essentially say, well,

how many combinations?

And that's the key word.

How many combinations are
there, where out of these

3 a plus b terms, I'm
choosing the a term?

I'm choosing 2 a terms.

Because to get a squared, I
have to pick an a term twice.

And so that's where I have
to pick 2 a terms out of

the three times I pick.

So I'm picking three times.

Two of the times, I'm
picking an a term.

So out of three times,
I'm choosing 2.

And that's where 3 choose
2 comes from, for the

a squared b term.

And so you could, for the ab
squared term, you could say,

well, I'm picking an a once.

How many ways are there to
choose a once when I'm

taking from 3 things?

So it could be 3 choose 1.

But that's also the same thing,
or should be equal to-- you

could also say, well, I'm
picking-- how many ways are

there to pick b twice if
I'm picking three times?

Well, if it's a b squared,
I'm picking b twice.

So that should be equal to
3 choose 2 ab squared.

And if you work these out, you
will find that, yes, these

both turn out to be 3.

And actually, that's why
there's some symmetry there,

and the combinations
all work out.

But hopefully, that's
giving you an intuition.

Essentially, when you're
doing the-- so the binomial

expansion of this--.

Let me just rewrite it again.

You know, this is a
plus b to the third.

It's 3 choose 0 of a cubed b to
the zero, plus 3 choose 1 of

a squared b to the
one, we could say.

Plus 3 choose 2 of ab squared,
plus 3 choose 3 of a

to the zero, b cubed.

So what's this saying?

What's 3 choose 3 saying?

How many ways-- if I'm choosing
from 3 different things-- how

many ways can I pick
exactly 3 b's?

That's how you can view it.

How many ways can I pick 3 b's?

I'm either picking
an a or a b, right?

You could say it's either a
heads or tails, or you know,

red or black or white,
but it's either a or b.

How many ways can I choose b
three times from 3 things?

Well, when you evaluate this,
you get this to b one,

and that makes sense.

Because it's 1b cubed.

And similarly, I mean, you
could view this as how many

ways, when I'm picking
out of 3 things, can

I pick exactly 0 b's?

That's 3 choose [? b. ?]

How many ways can I
pick exactly 0 b's?

Well, that's the same thing
as how many ways can

I pick exactly 3 a's?

This is also 1.

There's only 1 way to do it.

And that's the way.

Once again, there's only
1 way to do this one,

and this was the way.

There's 3 ways to
get a squared b.

There's 3 combinations.

There's 3 possible-- actually,
well, unique permutations.

But they're all the
same combination.

So there's 3 identical
combinations for picking 2 a's

and a b, and those are this
one, this one, and this one.

So hopefully, I
didn't confuse you.

And hopefully at minimum, that
gives you a glancing intuition

of why combinations are even
involved in the binomial

theorem, or whether they're
even involved when you're

expanding a binomial
to some power.

And at best, I really hope that
I've given you a deep intuition

for why this happens.

I will see you in
the next video.