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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.

  • So what is it about this that has so intrigued mathematicians the world over?

  • 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.

  • Or how about this: shade in all of the odd numbers.

  • 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

  • and get your answer.

  • 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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