# The mathematical secrets of Pascal’s triangle – Wajdi Mohamed Ratemi

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

(1×1) + (2×10) + (1×100). 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.

nobody take anything from this video for ms predas class, i will slap u

this is actually going to help me in things like combinations and probability neato

I might as well had watched that in Upper Wonga Wonga language…

杨辉三角形，可求多项式系数

👍👍

Balise Pascal may have communicated with extraterrestrials, if this topic was raised in HISTORY TV

Hmmm. What can we find next. Thats up to you

Can i pass matric first

feminism!

If this doesn't convince you that God is Math, nothing will.

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

my school has a poster about pascals triangle and it shows that if you start on the left edge, go down right along the diagonal for whatever amount then change direction for one then stop the number you stop at will be the sum of all the previous numbers you went through (I probably didn't explain that very well)

however someone looked closely then said it was wrong as it actually said you could start at any number but that isn't true and I figured out the mistake myself (BTW only one example they gave had this mistake)

omooo i just learned many tips from this video

thumbs upladies and gentleman, this video here is proof that illuminati invented maths.

I've been looking online for how come I found this pattern during my computer programming, each time had something to do with binary: one with refactoring a linked list to binary search efficiency, the other time also had something to do with binary but I forgot already. It is just amazing how frequently I find myself running into these patterns of numbers.

you can find more secrets of pascal triangle at this blog post

https://www.themaths.in/2019/05/pascals-triangle-hidden-secrets-and.html

1 2 3 4 5…

1 3 6 10 15…

1 4 10 20 35…

Thanks a lot✌️❤️

2:57 its the triforce

One beautiful property is following.

Take a lower Pascal Matrix like

1 0 0 0 0

1 1 0 0 0

1 2 1 0 0

1 3 3 1 0

1 4 6 4 1

And a corresponding upper Pascal Matrix

1 1 1 1 1

0 1 2 3 4

0 0 1 3 6

0 0 0 1 4

0 0 0 0 1

Multiply them.

What do u get? 🙂

3:10 TRIFORCE-CEPTION

1:15 (x+y)^5=………+5xy^4

I got the feeling this is the blueprint to space.

A kid in my algebra class wrote this on the board and the whole class looked at him like he was crazy

Did anyone see koch's curve?

Amazing video 👌^999!

Awe struck by watching the secrets of Pascal's triangle.

Thank you so much for enlightening us with such an awesome video.

Blown away by the way how one can generate the triangle fractal from Pascal's triangle 🤯!

Thank you once again 😊!

there is 1 more thing is discovered in this triangle but can't explain in comment.

Why did you not give hindi subtitle

In cc ?

Still Watching it in 2019

I m not lying but i actually discovered this sequence independently. When i was trying to generate a universal formula for calculating compound interest for cracking a govt exam. I have already remembered it upto five terms. And i can calculate compound interest in fractions of a second now. Didn't knew it was so important other than that.

Can you speak up?

Watched it for curiosity

Now I got a mathematics degree

TED-Ed Could you Explain 2:51 more Elaborately?

I thought It was called the Tartaglia Triangle (in my algebra exercise book)

this triangle was studyed also by Niccolo Fontana also know as Tartaglia.Infact it's called the Tartaglia's triangle.

The least impressive fact is the one about each being a power of 2… (x+y)^n and when n=1 the corresponding value is 2

Half of the patterns are already related to each other, though.

I love maths

i wish i know this when im still in school :'(

To the 5 dimensions we go!

Yeah nice

Ive also found a way to use it to find a^n – b^n using a summation and having rows of the Pascal triangle be coefficients

It isn't row 5, it is row 6.

Can someone change the sound of the introduction?

3:05 the greater triforce

Mean while in Indian Classrooms..

Find the 7th from the last co-efficient of x in the expansion (x+y) ^2019.

In YouTube : This.

Illuminati

🤯

Mindblown in the combinatronics part

Also, when summed up the elements of the Pascal's triangle's rows in two different ways they create the series of powes of 2 and 11 which are two amazing numbers in themselves. 2 being the smallest one digit prime number and 11 being the smallest 2 digit prime number. Also, 2 written in binary is 11. Math is just magical <3

I don’t understand but I love it.

On the last one. I think using combinatorics is easier when it comes to higher numbers rather than writing the whole pascals triangle. Imagine knowing how many groups of 5 you gonna have in 100 people. Just saying

You missed the Fibonacci sequence and the primes numbers in the triangle but good video anyway

I was amazed by how much can be done by a single triangle😲

For which we learn 20 formulas😤😢

Why 6th number of 12th row? Someone please clarify.

These guys absolutely destroyed that Hindi lol

3:05 you mean the triforce

I hate math

I'm lost

I don't belong here

3:05 I know it as the tri-force of Zelda

Mind blown !! 😍

IN CHEMISTRY also The Pascal’s triangle used to predict the ratio of heights of lines in a split NMR peak in NMR SPECTROSCOPY

Why am I watching this.

… in the fourth Line – 1-4-6-4-1 – ist Hidden the word 1-D-V-D-1 – its the TETRALOGIE of D-a-V-i-D or your Musik DVD with 4.7 GB.

What's funny is Pascal plagiarised the triangle from a Muslim mathematician.

This is the case with early Western scientific "discoveries" whether they admit it or not.

Ko

the Illuminati is strong in this video

In 7th grade right now and this is hella confusing I’m supposed to make an essay on this for my summer homework?

You lost me after 0:07

Thank you for exist ted-ed.

7 videos from you and my brain is on the verge of exploding

I'm having a math marathon right now

Learned this in computer science

back when the world was boring and when people gets bored they invented math!

The background music and the jumpy triangle all over the video charmed me more than the genius play of Pascal's triangle.

Amazing video!!! Thank you!

No wonder I love number

Calculus Students be like:

(x+y)ⁿ

you made a mistake in 1:15

Chave tabxom <3

4:03 I love how this guy says 12

Every row is a multiple of 11!

Row 1: 11

Row 2: 121

Row 3: 1331

…

Great❤️🎈

Didn’t Blase Pascal create the Pascaline?

what about pascals trihedron

Ganesh Pai who has eaten a pie told me about pi when I was eating a pie.

excellent video but please try as much as you can to go slower in the explanation because the voice off is going so fast

These numbers really relate to 11^n where n=any corresponding exponent

I Saw the triforce

The Pascal's triangle really helped a lot to improve in mathematics !

Guys here’s another pattern

Take row 2 and add the numbers

Now do it with row 3

Row 4

5

And so on

The pattern here is that its always 2 to the n

With n being the number of rows minus 1

Lets do row 6

1 + 5 + 10 + 10 +5 + 1 is 32

And 2 to the (6-1) is 2 to the 5 witch is 32

00:31 just like it always have been

Fans of Legend of Zelda at3:01:Coincidence? I think not!

Math is magical

My quadratic formula

https://twitter.com/tera_char/status/1205365799034204161?s=20

This is an error in your video, 1:15 . The fifth row, you repeated the 5(x^4)y. You missed the 5x(y^4)

This is a Tartaglia's Triangle, not the Pascal's Triangle!!

Nice

Blaise Pascal: Hey this is an amazing triangle! Look at all these patterns of math!

Illuminati:

Stay right where you are.Amazing…my fav subject ❤️❤️

K

no, its not pascal's triangle

it's OUR TRIANGLE