The mathematical secrets of Pascal’s triangle - Wajdi Mohamed Ratemi 

This is one of the best math videos I've seen all year--so many topics covered and explained well. I also enjoyed learning about the origins of Pascal's triangle in other cultures.

For (x+y)^n, use Pascal's triangle. For (x+y+z)^n, use Pascal's tetrahedron. For (x+y+z+w)^n, use Pascal's pentachoron (,which is in 4D)!

I had no idea how the Pascal's triangle was applicable to combinatorics, that's so cool!

Amazing! You've just missed the Fibonacci Sequence in this triangle.

Didn't understand half of this, still loved it.

i swear this is the same guy's voice but with 50 thousand different names

You can also use pascals triangel to check if a number is prime or not. Go to the n:th row and ignore the ones. If all the remaining numbers are devissible by n, then n is prime. If they are not all divissible, then n is not prime.

I was so fascinated with the pascal's triangle when I was 12 and saw a lot of shortcuts to math formulas. I showed it to my math teacher one day hoping it would help some of my classmates who found math to be difficult. She just dismissed the idea and said we better stick to the traditional formulas.

this guy's voice is so fucking nice

I had a math Prof where every year for his discrete math course he would just look at the Pascal triangle, make a pattern and design a problem around it to see who could make the connection

So the 5th row is in 4 dimensions

I enjoyed Pascal's triangle in 10th grade. I was finding a lot of really crazy patterns about the triangle before we learned about them as well as stuff in next year's course and I did pages of notes that were outside of the curriculum. I thought up a lot of theoretical stuff in the video and some crazy stuff that I forgot about. The school system doesn't like learning and thinking though, I thought I would impress my teacher by creating my own notes and problems that were beyond what we were learning to show my comprehension of it. I was finding ways of developing predictive patterns and structures beyond the Sirpinski Triangle. I did that instead of the assigned work, she disregarded everything and failed me. It killed my passion for math so hard that I dropped the course and took basic math courses to pass instead of AP, My once fiery passion for math is still dead to this day.

I'm in 9th grade and I'm crazy about Maths. This is the first time I actually heard about Pascal's tangle and I so desperately wanted to 'discover' a pattern on my own. I observed one after looking and thinking about it for 10 minutes, but something tells me this pattern has already been observed. Nevertheless, I'm quite proud of myself that I did this. If we look closely at the 4th diagonal-like/slant row, we can observe that the next alternate numbers in that row is the sum of the present number and the square of a natural number - the no. starts with 1 and keeps increasing for every alternate number. I'm not sure many of you understood what I told but I'll give an example : The 4th slant row is - 1,4,10,20,35,56,84,120,165,220 and so on. 10 = 1 + 3^2 20 = 4 + 4^2 35 = 10 + 5^2 56 = 20 + 6^2 84 = 35 + 7^2 120 = 56 + 8^2 165 = 84 + 9^2 220 = 120 + 10^2 ...............

I did not realize that I was assigned ASMR in my math class... this dude’s whispering and the rhythmic clicking made me sleepy

Honey : *i wanna have kids...* Me , an intellectual : *hold my Pascal's triangle*

They taught us at school that the Pascal’s triangle holds the answer to (x+y)^n but what they didn’t tell us was how amazing the triangle really is and how many answers it contains to so many mathematical problems. Thank you for bringing to my attention the amazingness that is the Pascal’s triangle.

Those snappy animations and musical points were just mesmerising

Great and very interesting video! Since I first learned about the Khayyam Triangle (I studied throughout undergrad with an Iranian professor and she taught me the habit of calling it Khayyam), I've found it to be one of the most beautiful and profound concepts in mathematics.

It's the Triforce!

I love these episodes! Keep up the good work!

Whenever I see Sierpinski's triangle, I can't stop my self from thinking about a tri-force made of tri-forces

Now I remember the poster of Pascal's Triangle in my High School math class. I never cared for it until now.

Wow, this is the first time i've ever seen the detailed and well-explained Pascal’s triangle topic, thanks u very much Ted-Ed

TED-Ed.....please add captioning or subtitles to all your videos! A number of deaf and hearing impaired audiences, including myself, would benefit from this so we do not get left behind.

I am inspired by his challenge in his last sentence: What might we find next? Well it's up to you.

Mathematicians are some of the smartest people in the world.

This just rose my interest in Permutation combination, binomial theorem and probability, really a great work.

i knew it, illuminati's invented math confirmed

My favorite pattern in Pascal's Triangle is that if you draw a diagonal line down the triangle, wherever you like, then at the row you stop, go one number in the opposite horizontal direction, and you will get the sum of all the numbers in the line you drew (not including the one in said row).

it's everyone's triangle okay geez

OMG, so amazing that Pascal's Triangle has such wonderful uses !!!

i knew about this triangle but had absolutely no idea that there were so many other applications! loved this video

*puts on Pascal's Triangle glasses* AAAH THE POINTED EDGES!

mind blown! I love math!! just wish I was better at it

this is so cool and I learned a lot!! Thanks, TEDed!!!

That's the thing I love about mathematics... Even the simplest of things can blow your mind away like anything. Cheers!

I like the geometric interpretation of a triangle in which the nth row corresponds to an n-dimensional cube. Row 0 is 1 (just a point). Row 1 is 1-1 (a line segment). Row 2 is 1-2-1 (a square, but a pair of points added on the diagonal). Row 3 is 1-3-3-1 (a cube; sliced into a point, two triangular vertex figures, and another point). This pattern continues through dimensions and actually generates n-dimensional polyhedra. Remember that diagonal that was the integers? In geometric terms, it's a point, a line, a triangle, a tetrahedron, a 5-cell, and so on.

- *But if you add thousands of rows, you get a fractal known as...* - TRIFORCE! -*Sierpinski triangle* - Oh

Thank you for exist ted-ed.

I LOVE THIS VIDEO ! I just love those math videos that show perfect animation (cute) instead of just scribbling on a whiteboard. Five stars to you!

We know that numbers in the Pascal's triangle corresponds to the coefficients of (x+y) raised to the power of the row number-aka the "binomial coefficients" (see 1:06). However, what's really amazing is that a recent paper has used this as a test for prime numbers. The test is simple: if all the coefficients that are more than 1 are divisible by the power, the power is prime. For example, raising to a power of 2 gives us: 1, 2, 1. 2 is divisible by 2. So 2 is prime. Raising to a power of 3 gives us: 1, 3, 3, 1. 3 is divisible by 3. So 3 is prime. Raising to a power of 4 gives us: 1, 4, 6 ,4 1. 6 is NOT divisible by 4. So 4 is not prime. Raising to a power of 5 gives us: 1, 5, 10, 5, 1. 5 and 10 are both divisible by 5. So 5 is prime. And the list goes on... ;) This test is known as the AKS primality test.

Modala video Appreciate your work 🎉 And this video is recommended by prestigious BRAND Mini University BHAVANS 👏👏🏆🏆

Was studying this last year (year 10) at school, (I took my statistics GCSE a year early), didn't quite understand it at the time but it makes a lot more sense now. Thanks!

this is amazing!!!! keep the good work!!

4:03 I love how this guy says 12

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 😊!

Thank you so much for uploading this video. It is helping me get through the pandemic!

The animations are just amazing !

Math is so beautiful that knowing its true beauty would cause stack overflow on the Universe and to terminate.

Pascal's triangle cal also be used for set theory, more specifically union sets nCr can yield two things. The first being the expansion equation for the union of infinitely many sets.That's, nCr specifically provides the alternative negative and positive intersect sets. Eg. (AUBUC) =A + B + C - (AnB + AnC + BnC) - AnBnC. .. which also lends itself to prob. theory. The second is, the P(AuBuC) = P(A) + P(B) + P(C) - P(AnB) - P(AnC) - P(BnC) + P(AnBnC). I'm still learning and working on the Statistical applications which uses Pascal's Triangle.

I didn't knew it had so many interpretations !Thank you.

If you insert fractions into the equations it still works and it’s like the numbers in between still add up correctly and you can even go through the triangle on a different side. It’s amazing

Before I die, I would dedicate a decade of my life just figuring out how many surprising things can this triangle reveal and I will demand a nobel prize if I ever discover one😩

Blaise Pascal: Hey this is an amazing triangle! Look at all these patterns of math! Illuminati: *Stay right where you are.*

I understood alot of knowledge in 5 mins ... I like it فهمت الكثير خلال فقط 5 دقائق عظيم !

Most of the thing I know about it but thanks very much for presenting this graphical video! Thanks a lot!

Video: known as the Sierpinski Triangle Me: Also known as what happens when the Triforce takes over.

Still Watching it in 2019

One of my favourite videos on internet!!

Thanks! I just find this channel, can't wait to share the videos to my students :)

I went to school and thought I knew Pascal's Triangle. Then I watched Numberphile and thought I knew Pascal's Triangle. Then I watched this. …And then I read the comments.

Math is love, math is life

I just figured out how to do the expansion by myself after watching this video, but I've spent a lot time in the past thinking about pascal's triangle, multiple dimensions, factorial, etc in an effort to find the numbers inbetween with an explanation that meets my satisfaction...

Loved your work thank you

2:53, and the next row would be how many you spheres you would need to make a 4D triangle

I take pride in being an Indian whenever it associates us among one of the first inventors/discovers of important scientific things. 😊

Absolutely superb animation.

I needed this for my Performance task in grade 8. Thanks! I learned a lot!

why these interesting topics are not told in school

4:05 I learned that in math class

everything you mentioned can be explained like this since it represents the binomial coefficients of (x+y) ^n 1) now let x=y=1 which makes it 2^n so if you add all numbers in a line you will end up with a number that can be represented as 2^n 2)let x=10 and y=1 =>every time x being raised it represents the decimal position while every time y is raised it just corresponds to 1 for example take (x+y)^2 = x^2 + 2xy + y^2 now take x=10 and y =1 (10+1)^2= 10^2 + 2*10*1 +1^2 now RHS can be written as 1(10^2) + 2(10^1) + 1(10^0) notice how the numbers in brackets represent the positional weights of our decimal system. now on LHS we have (10+1)^2=11^2 so that's why we always end up with a 11^n number 3) now the triangle can be explained like this to make a triangle with n rows (the arrangement should be as shown in the video) the number of spheres you need is sigma(n) or simply sum of n natural numbers for example for a triangle of 4 rows we need 4*(4+1)/2=10 spheres . since if we check the roots of each number inte 3rd diagonal we will find out that we are summing up the n natural numbers in this case n=position of the number in the diagonal 4) now for the part of the tetrahedron we can imagine this as the above traingles being arranged on one another in order that is always the triangle below nth triangle ( triangle of n rows) will be (n+1)th triangle (triangle of n+1 rows) we can now think this as summing up of sum of n numbers to which we can derive the formula as number of spheres reqired = 1+3+6+....+sigma(n) and if we trace back to the origins of each number in the 4th diagonal and rearrange them we will end up with such series... 5) now since we know that each row in the triangle is related to the binomial coefficients of (x+y)^n relating to binomial theorem you can tell that each number is nothing but (n-1)C(z-1) where n is the row and z is its position in that row and that is what we do to find combinations and with a little bit of more for "probability" i hope this helps someone to understand better

I needed this video!!

0:21 "in China it's John Wayne's triangle"

oh look my triangle!:3

Brilliantly animated!

This might be one of the most enlightening videos of Teded

The property with eleven is kind of redundant. That is just taking the expansion of (x+y)^n and replacing for x = 10 and y = 1

Every row is a multiple of 11! Row 1: 11 Row 2: 121 Row 3: 1331 ...

I love this triangle, by whatever name you choose to call it. I had to do a research project and presentation on it in my theoretical math class last semester, and I kinda wish I had seen this video then.

-Thanks that was very helpful for me to know since I'm doing pascal is school! -I'm going to like this video!

Reminds me of Vihart lol

In Italian we say "Triangolo di Tartaglia"

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.

Glad I watched this. I couldn't remember what it was that I used Pascal's triangle for in my Discrete Mathematics class but this finally reminded me that it was Combinatorics.

I shall show this video to everyone who says math is boring

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

Amazing video!!! Thank you!

That was a great breakdown! I definitely needed it. Because I have no idea what is happening in class with it! lol

And this video is recommended by prestigious BRAND Mini University BHAVANS 👏👏🏆🏆

I use the pascal's triangle in EPR spectrums, when I have the same coupling and S=1/2, I can predict the relative intersity using it.

Very nicely done.

When you showed me Sierpenski's Triangle, I saw the Triforce from Zelda

this really works for me , had no idea about pascals triangle thanks to this guy

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

Pascal's triangle please notice me!!!

I DONT now this triangle earlier and do it on a paper, then i watch this video. 😂 I invented it to.

thank you for such nice videos

I don’t know a triangle with numbers can be this awesome!

Triforce

0:16 Why you don't mention the year in which Pingala discovered it..Don't be too smart...Let everyone know that it was discovered by Bharat (India)