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This may look like a neatly arranged
stack of numbers,
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but it's actually
a mathematical treasure trove.
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Indian mathematicians called it
the Staircase of Mount Meru.
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In Iran, it's the Khayyam Triangle.
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And in China, it's Yang Hui's Triangle.
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To much of the Western world,
it's known as Pascal's Triangle
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after French mathematician Blaise Pascal,
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which seems a bit unfair
since he was clearly late to the party,
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but he still had a lot to contribute.
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So what is it about this that has so
intrigued mathematicians the world over?
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In short,
it's full of patterns and secrets.
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First and foremost, there's the pattern
that generates it.
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Start with one and imagine invisible
zeros on either side of it.
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Add them together in pairs,
and you'll generate the next row.
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Now, do that again and again.
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Keep going and you'll wind up
with something like this,
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though really Pascal's Triangle
goes on infinitely.
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Now, each row corresponds to what's called
the coefficients of a binomial expansion
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of the form (x+y)^n,
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where n is the number of the row,
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and we start counting from zero.
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So if you make n=2 and expand it,
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you get (x^2) + 2xy + (y^2).
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The coefficients,
or numbers in front of the variables,
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are the same as the numbers in that row
of Pascal's Triangle.
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You'll see the same thing with n=3,
which expands to this.
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So the triangle is a quick and easy way
to look up all of these coefficients.
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But there's much more.
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For example, add up
the numbers in each row,
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and you'll get successive powers of two.
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Or in a given row, treat each number
as part of a decimal expansion.
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In other words, row two is
(1x1) + (2x10) + (1x100).
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You get 121, which is 11^2.
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And take a look at what happens
when you do the same thing to row six.
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It adds up to 1,771,561,
which is 11^6, and so on.
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There are also geometric applications.
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Look at the diagonals.
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The first two aren't very interesting:
all ones, and then the positive integers,
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also known as natural numbers.
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But the numbers in the next diagonal
are called the triangular numbers
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because if you take that many dots,
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you can stack them
into equilateral triangles.
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The next diagonal
has the tetrahedral numbers
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because similarly, you can stack
that many spheres into tetrahedra.
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Or how about this:
shade in all of the odd numbers.
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It doesn't look like much
when the triangle's small,
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but if you add thousands of rows,
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you get a fractal
known as Sierpinski's Triangle.
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This triangle isn't just
a mathematical work of art.
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It's also quite useful,
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especially when it comes
to probability and calculations
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in the domain of combinatorics.
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Say you want to have five children,
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and would like to know the probability
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of having your dream family
of three girls and two boys.
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In the binomial expansion,
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that corresponds
to girl plus boy to the fifth power.
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So we look at the row five,
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where the first number
corresponds to five girls,
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and the last corresponds to five boys.
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The third number
is what we're looking for.
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Ten out of the sum
of all the possibilities in the row.
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so 10/32, or 31.25%.
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Or, if you're randomly
picking a five-player basketball team
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out of a group of twelve friends,
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how many possible groups
of five are there?
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In combinatoric terms, this problem would
be phrased as twelve choose five,
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and could be calculated with this formula,
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or you could just look at the sixth
element of row twelve on the triangle
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and get your answer.
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The patterns in Pascal's Triangle
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are a testament to the elegantly
interwoven fabric of mathematics.
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And it's still revealing fresh secrets
to this day.
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For example, mathematicians recently
discovered a way to expand it
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to these kinds of polynomials.
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What might we find next?
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Well, that's up to you.