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PROFESSOR: We're going to spend
a couple of minutes talking

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about the binomial
theorem, which is probably

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familiar to you
from high school,

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and is a nice first
illustration of the connection

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between algebra and computation.

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So the idea that
underlies the connection

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is illustrated by
the distributive law.

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And I'm purposely writing
it in this wacky way--

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it's purely symbolic--
where I'm going to multiply

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three beanies by two neckties.

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What do I get if I multiply
three beanies by two neckties?

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Well, applying the
distributive law

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says that we get every
possible pairing of a beanie

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and a necktie
multiplied together

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and you add them all up.

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So there are three terms
here and two terms there.

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I'm going to wind up with six
terms, which you see laid out.

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The basic rule is that a product
of sums by the distributive law

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becomes a sum of products.

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And the sums that
you get involves

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products of all of the terms
from each of the components

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in every possible way.

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Let's look at that as it
applies to the binomial theorem.

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So the binomial theorem is
interested in the question

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of let's look at
the expression 1

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plus x raised to the NTH power.

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And we know that this will
be a polynomial of degree n,

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so it can be written in the form
a constant, c0 plus c1 times

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x to 1, c2 x to the
2, cn x to the n.

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And we like to ask, what
are the expressions for ck?

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So for example, here's a layout
of the first four powers of x.

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Let's say 1 plus x to the fourth
is 1 plus 4x plus 6x squared

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4x cubed 1x fourth.

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What's the pattern that
underlies those coefficients,

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1, 4, 6, 4, 1?

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Well, one way to
think about it is

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that if I wrote out
1 plus x to the n

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fully it is, of course, a
product of n occurrences of 1

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plus x times 1 plus x.

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And applying the distributive
law, the product of sums

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equals sum of products rule, but
this time there are n products.

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And what I wind up with is 2 to
the n terms that I'm adding up.

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Each of them is a product of
a selection of n items, one

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from each of the factors.

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So a term among these
2 to the n terms

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corresponds to
selecting a 1 or an x

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from each of the n factors.

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So if I started off, for
example, by selecting a 1,

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a 1, a 1 from each
of the n factors,

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I'd get the term 1 to the n.

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If I selected an x, an x, an
x, an x from each of the terms,

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I'd get this last
term x to the n.

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And if I made some arbitrary
selection, like I selected

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an x from the first one, and a
1 from the second one, and a 1

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from the third one-- I'm reading
this term-- 1 from the fourth,

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and so on, and x from the
next to the last, and a 1

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from the last, I would get this
particular term, which is not

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the next one that would
occur in alphabetical order,

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but it's just an example.

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So that's the simple idea.

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If you multiply out n
terms, each of which

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is a sum of two
things, you're going

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to wind up with 2 to the
n terms corresponding

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to every possible
way of selecting

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one or the other
of the components

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from each of the n products.

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So what's the coefficient
of x to the k?

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Well, the coefficient
of x to the k

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is the number of those
terms among the 2

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to the n in which
the power of x is k.

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That is, in which I selected
k x's and willy nilly

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n minus k ones.

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Well, how many ways
are there to select

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k x's among these n terms?

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And we know the answer to that.

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It's n choose k.

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It's all the ways of
choosing a subset of k items

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out of n items.

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And that's the answer.

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ck is simply n choose k.

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And this is called
the binomial formula.

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So now we know that
1 plus x to the n

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is n choose 0 plus n choose
1x, n choose 2x squared.

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n choose kx to the k
is the general term.

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Ending with n choose
n to the x to the n.

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So this expression, 1 plus x,
is called a binomial expression.

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And the choose numbers,
which we've seen previously,

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is the number of ways
to choose, in this case,

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k out of n elements, are
called binomial coefficients.

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And this is why they're
called binomial coefficients.

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So if I was going to
express it more generally,

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I didn't need it to be 1.

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I used 1 plus x just
because it was easier

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to follow the structure
of the formula.

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But if I look at x plus
y to the NTH power,

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again, the coefficients
are the same.

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It's n choose 0, but this
time y to the n, n choose 1,

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xy to the n minus 1.

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What's happening now is that
I'm choosing an x or a y

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instead of an x or a 1
from each of the terms,

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so that the xy terms are always
going to have a sum of degrees

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that is equal to n.

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If I choose k x's, I must
have chosen n minus k y's.

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And there it is
expressed in more concise

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form using sigma notation.

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That is the binomial formula.