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.