Binomial Theorem (part 1)

Introduction to raising (a+b)^n

We'll now learn an application
of combinations that you

probably won't find
intuitive at first.

But the more you think about
it, it will make a lot of sense

and hopefully it'll make you
appreciate, once again, the

beauty of mathematics.

And then you'll also know why--
when we say n choose k in

combinations --why that's also
called a binomial coefficient.

Because we are going to
cover the binomial theorem.

So before I give you the
binomial theorem, let's

understand the motivation for
why it's even interesting.

So let me erase that.

Invert colors.

So if we just had to
multiply-- I don't know.

Well let's just take different
powers of a binomial.

A binomial is just a polynomial
with two terms, right?

So a plus b-- Well a
plus b to the 0, that's

equal to 1, right?

Anything to the 0
is equal to 1.

a plus b to the 1, well
that equals a plus b.

a plus b squared, that equals--
and if you don't have a lot of

practice doing this you
might be tempted to say a

squared plus b squared.

But you should correct yourself
quickly and slap yourself on

the wrist or the brain or
someplace if you did that.

Because that's a plus
b times a plus b.

And then we could use the
distributive property, or,

if you learned it this way
in Algebra I, you could

use the FOIL property.

That equals a times
a plus b, right?

Plus b times a plus b.

Which is equal to a squared
plus ab plus ba plus b squared.

Which is equal to a squared
plus 2ab plus b squared.

That should be a bit
of review for you.

And now it gets a little
bit more interesting.

What is-- Let me circle that
just so we remember it.

That's a plus b squared.

What's a plus b to the third?

And now this is starting
to get complicated.

This is equal to a plus b times
a plus b times a plus b.

Or another way to view it,
it's a plus b squared

times a plus b, right?

This is a power of three.

So this was a plus b squared.

So if we multiply it by
a plus b, we'll get a

plus b to the third.

So let's do that.

Let's multiply this
times a plus b.

So first let's multiply
everything times b.

So that's b-- let me do
this in another color.

a squared b, right?

That's a squared times b.

Now let's do 2ab times b.

So plus 2ab squared, right?

2ab times b.

And then plus b cubed.

And then we have a times a.

Well that's a cubed, right?

None of these match that, so
I'll put it in another column.

a times 2ab.

Well that's 2a squared b.

I'll put that out
here: 2a squared b.

And then a times b squared.

Well that's plus ab
squared, right?

And now we'll just add
up all of the terms.

All we do is the distributive
property again, right?

We multiplied a times all of
these terms and then added that

to b times all of these terms.

If we add it all up-- I'll
try to do it in order

of-- Let's see.

Let's put the a cubed first.

And then-- Well actually we
already had this thing.

This 2a squared b, I could
have written it here.

2a squared b, because I had
an a squared b here so.

I just rewrote the
2a squared b here.

So we have a cubed plus 2a
squared b plus a squared b.

That's 3a squared b.

And then 2ab squared
plus ab squared.

That's 2ab squared.

And then plus b cubed.

As you can see, that involved
a lot just to take something

to the third power.

So we could-- If we had the
time, we could figure out what

a plus b to the fourth power is
or what a plus b to

the tenth power is.

But as you could imagine,
this would take you all day.

So wouldn't it be neat if there
were an easy way to calculate

what a binomial is to
an arbitrary power?

And that's where the binomial
theorem comes into play.

And in this video I'm going
to show you what the

binomial theorem is.

I will show you
how to apply it.

I will show you a trick or
a technique that will make

you seem like a genius.

And then in the next video
I'll hopefully give you

some intuition for why the
binomial theorem actually

involves combinations.

Why it involves actually the
binomial coefficient at all.

So what is the
binomial theorem?

Let me erase all of this.

And you can confirm that the
binomial theorem works for the

ones that we've worked out,
up to a plus b to the third.

You could work out a plus
b to the fourth if you

like to punish yourself.

Let's see.

Clear image.

Invert color.

So the binomial theorem tells
us that a plus b to the nth

power is equal to-- And I know
it's going to look complicated

at first, but we'll do a couple
of examples and you'll see

it's not that intimidating.

It equals the sum from k
equals 0 to n, right?

This n is the same
thing as that n.

Of-- each term is n
choose k, right?

We're going to keep
incrementing k up from 0 to

n-- of x to the n minus
k times y to the k.

I know that looks complicated
but if we do a couple of

concrete examples, I think it
should make a reasonable

amount of sense.

So given-- Oh sorry.

This is-- This isn't-- I
was copying this down.

This should be a to the n
minus k and this should be b.

What I had written down
before, that would be

x plus y to the n.

If we have a plus b to the n, n
choose k each term. a to the

n minus k times b to the k.

So let's apply this, a couple
of concrete examples.

We could even switch around the
variable names if we want just

to show you that they don't
have to be a's and b's.

They can be anything.

So what is a plus b-- let's do
one that we otherwise would

have found fairly difficult-- a
plus b to the fourth power.

Well that, the binomial theorem
tells us that, let's see, the

first term is going to be--
Well what's n, first of

all? n is 4 in this case.

It equals-- Let me fill in
all the numbers actually.

From k equals 0 to 4
of 4 choose k, right?

Because k is what
we're incrementing.

a to the 4 minus k.

b to the k, right?

I just substituted the
n into the binomial

theorem definition.

And what does that equal?

Well the first term
is k equals 0.

So that's 4 choose 0.

So out of 4 things, I'm
going to choose 0.

And I'll show you in the
next video why that works.

Of a to the 4 minus k.

Well the first term k is 0.

So it's a to the fourth,
b to the 0, right?

So the b, that's just 1,
so we can just ignore it.

So what's the next term?

Well, it's going
to be 4 choose 1.

And now k is 1.

So 4 minus 1 is 3.

a to the third.

And k is 1 now.

We're in the-- This
is the zeroth term.

This is the first term.

So b to the first plus-- So as
you can see, each term we go,

the a term, the first term,
whichever it is, it decrements.

It starts up at the power
n, or in the fourth power.

And then each term,
it goes down by 1.

And then the second term,
the b term, it starts

at the zeroth power.

So it starts at 1.

That's why you don't
see it there.

And then each term
it increments up.

So then the next one-- So I
think you see the pattern.

It's going to be 4 choose 2, a
squared b squared plus 4 choose

3 ab to the third
plus 4 choose 4.

Now it'll have a to the
zero, so that's just

1, b to the fourth.

So we're done if we just figure
out what these binomial

coefficients are.

And that's where they
come from, from the

binomial theorem.

But we remember how to
calculate that, right?

In general-- and hopefully you
have the intuition on this.

You shouldn't just memorize it.

n choose k from our
combinatorics is equal to n

factorial over k factorial
divided by n minus k factorial.

So in this case,
what's 4 choose 0?

That equals-- I know it seems
very time consuming right now.

Although it's less time
consuming than actually

multiplying it out.

But I'll show you trick in a
second that will amaze you.

So this is equal to 4 factorial
over 0 factorial times 4

factorial, right-- 4 minus 0 is
4 --a to the fourth plus 4

factorial over 1 factorial
times 3 factorial, right?

4 minus 1 is 3 factorial.

a to the third b plus-- I know
this is getting a little

tedious, but I think it's good
to completely work through

one entire problem
--plus 4 choose 2.

That's 4 factorial over
2 factorial times 2

factorial, right?

4 minus 2 is 2.

a squared b squared
plus 4 choose 3.

That's 4 factorial
over 3 factorial.

4 minus 3 is 1 factorial.

ab cubed.

And then 4 choose 4.

That's plus 4 factorial
over 4 factorial times 0

factorial b to the fourth.

And notice: this coefficient is
the same as that coefficient.

This coeffieicnt is the same
as this coefficient and then

this one's in the middle.

So let's evaluate them.

And I'll switch colors.

So 0 factorial, in case
you don't know it, it's

actually defined to be 1.

Which is a little bit
non-intuitive because 1

factorial is also 1.

But that's just something
you should know.

So 4 factorial divided by 0
factorial times 4 factorial.

This is actually equal to 1.

So the first term is just a to
the fourth plus 4 factorial--

it's 4 times 3 times 2 times 1
--divided by 3 times 2 times 1.

So that equals 4.

4a cubed b plus 4 factorial.

That's 4 times 3
times 2 times 1.

So that's 24, right?

Over-- what's 2 factorial?

That's just 2.

So 2 times 2 is 4.

So 24 divided by 4 is 6.

So 6a squared b squared plus
well 4-- This term is the

same as this term, right?

With just the 1 factorial
and the 3 factorial

got switched around.

And you might want to think
about that for a few

seconds as to why that is.

It should make a
little sense to you.

But that is-- So it's
going to be 4ab cubed.

And it makes sense, right?

Because this could have
just been b plus a.

A plus b and b plus a are the
same thing so it makes sense

that there's a symmetry, right?

That we have 4ab cubed and
we also have 4a cubed b.

Ignore me if you find
that confusing.

If you find it enlightening,
all the better.

And then the last term.

4 factorial.

This term is the same
thing as this term.

And we've already figured
out that equals 1.

So plus b to the fourth.

So I had a little symmetry.

The coefficients
are 1, 4, 6, 4, 1.

And I'll show you in a future
video that these are actually

the terms of a Pascal Triangle,
which is another avenue

to go in mathematics.

But anyway, this was
an application of the

binomial theorem.

And I realize I've taken
12 minutes so far.

So I will do more examples
in the next video.

See you soon.