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• This may look like a neatly arranged stack of numbers,

• but it's actually a mathematical treasure trove.

• Indian mathematicians called it the Staircase of Mount Meru.

• In Iran, it's the Khayyam Triangle.

• And in China, it's Yang Hui's Triangle.

• To much of the Western world, it's known as Pascal's Triangle

• after French mathematician Blaise Pascal,

• which seems a bit unfair since he was clearly late to the party,

• but he still had a lot to contribute.

• In short, it's full of patterns and secrets.

• First and foremost, there's the pattern that generates it.

• Start with one and imagine invisible zeros on either side of it.

• Add them together in pairs, and you'll generate the next row.

• Now, do that again and again.

• Keep going and you'll wind up with something like this,

• though really Pascal's Triangle goes on infinitely.

• Now, each row corresponds to what's called the coefficients of a binomial expansion

• of the form (x+y)^n,

• where n is the number of the row,

• and we start counting from zero.

• So if you make n=2 and expand it,

• you get (x^2) + 2xy + (y^2).

• The coefficients, or numbers in front of the variables,

• are the same as the numbers in that row of Pascal's Triangle.

• You'll see the same thing with n=3, which expands to this.

• So the triangle is a quick and easy way to look up all of these coefficients.

• But there's much more.

• For example, add up the numbers in each row,

• and you'll get successive powers of two.

• Or in a given row, treat each number as part of a decimal expansion.

• In other words, row two is (1x1) + (2x10) + (1x100).

• You get 121, which is 11^2.

• And take a look at what happens when you do the same thing to row six.

• It adds up to 1,771,561, which is 11^6, and so on.

• There are also geometric applications.

• Look at the diagonals.

• The first two aren't very interesting: all ones, and then the positive integers,

• also known as natural numbers.

• But the numbers in the next diagonal are called the triangular numbers

• because if you take that many dots,

• you can stack them into equilateral triangles.

• The next diagonal has the tetrahedral numbers

• because similarly, you can stack that many spheres into tetrahedra.

• It doesn't look like much when the triangle's small,

• but if you add thousands of rows,

• you get a fractal known as Sierpinski's Triangle.

• This triangle isn't just a mathematical work of art.

• It's also quite useful,

• especially when it comes to probability and calculations

• in the domain of combinatorics.

• Say you want to have five children,

• and would like to know the probability

• of having your dream family of three girls and two boys.

• In the binomial expansion,

• that corresponds to girl plus boy to the fifth power.

• So we look at the row five,

• where the first number corresponds to five girls,

• and the last corresponds to five boys.

• The third number is what we're looking for.

• Ten out of the sum of all the possibilities in the row.

• so 10/32, or 31.25%.

• Or, if you're randomly picking a five-player basketball team

• out of a group of twelve friends,

• how many possible groups of five are there?

• In combinatoric terms, this problem would be phrased as twelve choose five,

• and could be calculated with this formula,

• or you could just look at the sixth element of row twelve on the triangle

• The patterns in Pascal's Triangle

• are a testament to the elegantly interwoven fabric of mathematics.

• And it's still revealing fresh secrets to this day.

• For example, mathematicians recently discovered a way to expand it

• to these kinds of polynomials.

• What might we find next?

• Well, that's up to you.

This may look like a neatly arranged stack of numbers,

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B1 US TED-Ed triangle pascal expansion mathematical stack

# 【TED-Ed】The mathematical secrets of Pascal’s triangle - Wajdi Mohamed Ratemi

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SylviaQQ posted on 2016/06/30
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