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

View full lesson: ed.ted.com/lessons/the-mathema...
Pascal’s triangle, which at first may just look like a neatly arranged stack of numbers, is actually a mathematical treasure trove. But what about it has so intrigued mathematicians the world over? Wajdi Mohamed Ratemi shows how Pascal's triangle is full of patterns and secrets.
Lesson by Wajdi Mohamed Ratemi, animation by Henrik Malmgren.

[@MindYourDecisions]

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.

[@youssofrabie5236]

+MindYourDecisions point out the explaination, please! cause I thought he was just telling stuff in the video but first let's try to agree on a definition to the term explaination. propose one, hence I'll look for explaination in the videos, or pick the easier way of course if you have time to do so. You're understandable if not!

[@FD1CE]

+MindYourDecisions, There exists much much much more properties of this magical triangle

[@MrDamojak]

Gregorio Salazar When you are ready keep watching the video.

[@alexpagnetti585]

hey it's presh talwalker

[@ezramahle1472]

haha

[@MatematicaRio]

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

[@loriefranceschi2590]

+Matemática Rio And the Golden Ratio using Fibonacci Sequence in the Pascal's Triangle.

[@MatematicaRio]

+Rafael Moraes tenho um que fala do triângulo de Pascal e outro de Sierpinski. Pode procurar aí.

[@loriefranceschi2590]

+Rafael Moraes the video does mention other mathematicians who came up with the triangle. In most of the Western world we call it Pascal's Triangle.

[@luccamarcal4119]

Nossa n esperava vc aqui, vc é um mito

[@denelson83]

And the Chebyshev polynomials.

[@nayutaito9421]

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)!

[@leosong829]

Wtf

[@louiswong921]

cool

[@Navvye]

Man, did that guy love expansions!

[@leosong829]

@QuantumRat2005 oh i understand that. but wtf is pascals pentachoron

[@andreaslind6338]

@@leosong829 it's a four dimensional solid, that gives numbers in 4 dimensional space, just as Pascal triangles gives numbers in 2 dimensional space.

[@AnstonMusic]

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

[@AnstonMusic]

***** You might wanna check the title of this video. Besides, who cares as long as we're talking about the same thing?

[@xxuncexx]

+Anston Music honestly who gives a flying feather?

[@AccuphaseMan]

+Emperor Theodor Tronét I'm persian, so this is probably bias, but i agree

[@AccuphaseMan]

+Anston Music Why is it either dashie or fluttershy?

[@AnstonMusic]

Dean Vidafar Uhh, I really don't know, I think I'm going to switch it around soon enough, I don't have a "best pony".

[@Nightsearch]

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

[@adflicto1]

+Nightsearch xD

[@thelightcrystals8332]

same

[@miaa7097]

u didn't get abused by math like I did lol

[@ramprasath219]

You can watch it once more. Because I understood it while watching it for the first time. You should also listen, not just watch!

[@weebghurl790]

Fr HAHAHA

[@Verlisify]

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.

[@Danieleghedix]

+Verlisify that's such a shame though...such a sad story..teachers play such a huge role in everyone's future passions and aspirations and very few people realize that.

[@Verlisify]

+Daniele Ghedalia absolutely, some teachers I really liked and kept with that. I took multiple culinary classes because my cooking teacher was great and made it fun to learn life skills and was ok with experimenting in recipe deviations. I brought in a recipe I made at home and it became something we made in the next week!

[@shweet7891]

read the deliberate dumbing down of America you're right, it's free pdf.

[@simoncarlile5190]

+Verlisify I feel ya, man. In seventh grade my teacher told our class to use 22/7 for pi. I entered it into a calculator and saw that it had a repeating pattern, but I happened to know that pi doesn't repeat. I asked her and she said that 22/7 was "basically pi". So I did a little research and discovered 377/120 was more accurate. I used that on a test and missed every question involving pi. When I talked to her and pointed out my value gave a better answer, she said I was wrong solely because I hadn't done what she told me to do. So in high school and beyond, I do just enough in class to pass, but I'm always writing down ideas (series of numbers, weird ways to generate constants, etc.) in my own notebook. Fuck formal education; learn math or anything else that interests you in your own way, and don't give up on it because some asshole tries to control how you think.

[@Verlisify]

+Simon Carlile Wow, thats upsetting. I have had my run in with stupid teachers, its about the answer key and not properly teaching people anything

[@oskarlindelof9685]

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.

[@cubicardi8011]

oskar johansson nice!!!

[@pedronunes3063]

oskar johansson Nice one.

[@pedronunes3063]

@@adam-markgheczi4577 No, row 3 is 1 3 3 1. 1 2 1 is row 2 and works as well

[@pedronunes3063]

@@adam-markgheczi4577 The first 1 alone is the row 0.

[@imrannaseer3095]

I was looking for such a comment.thnx bro

[@RomanNumural9]

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

[@SkinsFirstGeneration]

this guy's voice is so fucking nice

[@delaneysutherland5432]

thanks

[@allisonhalgreen4248]

the fact that u made it sound so weird🤣

[@omegasys8863]

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

[@harkrits.nagpal6505]

Omegasys it's Addison Anderson

[@sabriswish]

dudes voice makes me wanna clear my throat

[@aliaj.shafira7203]

He is only the narrator, not a person who givea the lesson

[@johnsonlawalgm]

No the person who created the lesson is the name that you are seeing but the voice name isnt shown

[@aliaj.shafira7203]

@@johnsonlawalgm I mean this one

[@ariztrad]

So the 5th row is in 4 dimensions

[@TheRedstoneTaco]

EXACTLY

[@krisnaga91]

my fucking mind is blewn

[@TheRedstoneTaco]

I can use a formula I created to go into infinite spacial dimensions. It took me a whole week to solve it but I was so glad when I did it.

[@mistakesweremade58]

What is infinite special dimensions?

[@TheRedstoneTaco]

A flat plane in space is the second spatial dimesnion. A line is the first spatial dimension. I am talking about a special set of polygons called triangular-based polygons formed in all of the infinite amount of spatial dimensions. Like the fourth spatial dimension, fifth spatial dimension, sixth, seventh, etc. I made a formula concerning them.

[@nikulitroblox]

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.

[@alejandronavarro4128]

Thats good thinking tho. Use every tool at your disposal. Work smart, then work hard.

[@silentoccasion4359]

Gemnicole just dont tell the teacher that you are using them

[@Jess-fo8sp]

I know that feeling >w

[@Brahvim]

School hates us. Thanks.

[@DhrithionVocals]

I hate it when you still get the correct answer using a different and more efficient method but the teacher says you have to do it HER way...

[@ra_mry3062]

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

[@luksum7128]

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.

[@Zahid_Abbas]

Cartesian Coordinate System in Detail: kzbin.info/www/bejne/bJXPh6iiaZaKmLc

[@samithhegde1753]

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 ...............

[@cheongziyong8871]

Congratulations, you've just rediscovered the tetrahedral numbers. How about a challenge? Find the rule that dictates the sequence of numbers that run exactly down the middle of the triangle (1,2,6,20...).

[@samithhegde1753]

+Cheong Ziyong They're called tetrahedral numbers? Sorry, I didn't know about them. And it's been well over 20 minutes, I've been trying to find a pattern, but I'm not able to. I searched in Wikipedia too, but I couldn't understand a thing over there. Something about binomial coefficients. Could you please explain it to me and tell me what the pattern is, in simple terms?

[@cheongziyong8871]

They are basically the central coefficient (the number in the front) of the expansion of (x+y)ⁿ. For example, (x+y)²=...+2xy..., (x+y)⁴=....+6xy.... and so on. You can generate them using 1/√(1-4x)

[@samithhegde1753]

+Cheong Ziyong I understood the first part, but what do you mean it can be generated using 1/ _/^(1-4x) ?

[@cheongziyong8871]

1/√(1-4x) is simply the generating function. It means that 1/√(1-4x)=1+2x+6x²+20x³+...

[@TheRedstoneTaco]

Those snappy animations and musical points were just mesmerising

[@IndigoIceman]

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.

[@ShawnRavenfire]

It's the Triforce!

[@sidneytan1615]

I c u in every TEDED vid doe lol

[@vboy360]

+Shawn Ravenfire lol XD

[@slimslimslimslimslim]

Teghsbshehejdyekdteje its a try force

[@JaytleBee]

+Shawn Ravenfire EVERYONE says that! It's so annoying! It's not a religious object from a video game, it's one of the most awesome fractions ever!

[@ducc5198]

No it's a fractal and a really amazing set of numbers

[@eredin2376]

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

[@shay3355]

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

[@BludMonger]

Hold my maths 😂😂😂

[@Perririri]

Normie

[@Laittth]

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

[@nagachika.g]

Parents didn't force us to learn maths for nothing. I regret not pushing my parents to force me to have a passion for maths when I was a kid. Maths truely is the greatest thing ever.

[@mli3793]

I love these episodes! Keep up the good work!

[@kentjohnduga6833]

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

[@anmol3]

Primes

[@arcadioleon7048]

pascals trihedron (3d triangle)

[@dilbert2182]

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

[@delaneysutherland5432]

hey dilbert

[@sonxuannguyen1207]

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

[@avinashyadav8314]

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

[@mikeblack9109]

Mathematicians are some of the smartest people in the world.

[@mikeblack9109]

***** wat

[@anmol3]

Physicists are too. Though I like maths more. 😂

[@simran210]

well nah durrrrrrrrrh

[@leosong829]

Yes

[@hiteshaithani2618]

@Jeff Sklownski thanks you i am also a mathematician

[@dawsy1000]

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).

[@nicoleeeeeeeeebiebs]

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

[@AmazingGriffin]

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!

[@danki2000daniel]

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

[@RTillero70]

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.

[@camilleabby6022]

Rebecca Grant they got subtitles

[@anujarora0]

Rebecca Tillero turn on the caption.When you see the videos on KZbin there are three dots on the right side corner.Click on the dots and turn on the captions and God bless you😊😊

[@cortneymaitland4429]

You're not missing much

[@Mxnh]

@@cortneymaitland4429 yes he is

[@whycantihaveaproperusernam9384]

Me, seeing this comment after KZbin removed community captions:

[@Goldslate73]

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

[@ThiNguyen-il9im]

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

[@kentlab3850]

i knew it, illuminati's invented math confirmed

[@ireallyhatemakingupnamesfo1758]

No, the illuminati is math, there's no denying, the illuminati controls the entire fabric of reality, both abstract and concrete

[@MouseGoat]

+I really hate making up names for this kind of stuff so illuminati = god, this anser so much but rase som rather interesting questions.

[@shweet7891]

illuminati here, we didn't invent math, we have a secret. triangle is composed of nothing but mathematics. no conspiracy theories though about being pop stars and secret rulers and lizards. that's just propaganda to destroy our efforts to reveal the secret

[@shweet7891]

+banana aya playa illuminati are not satanic 666 religious cases

[@elliotville7820]

Get out 👉🚪

[@Holobrine]

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.

[@kingshark422]

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

[@Meow_yj]

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

[@DeceitNLie]

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

[@KINGKOHIRR]

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

[@xploreourterra6096]

Jai BS 🤡

[@floydnelson92]

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...

[@rogersledz6793]

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

[@wrenk]

4:03 I love how this guy says 12

[@lake4461]

it's everyone's triangle okay geez

[@typo691]

**Our triangle** (cue USSR anthem)

[@KanishkMalkan]

Hahaha 😂 😂 😂

[@WillG_MTB]

@Think Is Awesome no

[@sohamshah5572]

This has 121 likes, imma keep it that way

[@WeCodeCurious]

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!

[@cattidesjar4229]

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

[@TerryMun]

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.

[@AdhiNarayananYR]

+Terry Mun This is amazing :)

[@TerryMun]

***** Might be known, but not proven until recently.

[@TerryMun]

That has been proven, yes, but what the AKS test is the first one that satisfies all of these: polynomial, deterministic, unconditional and global :) it leverages on the combination components logic, but the original intuition you have mentioned was not proven as useful tool to check of primality until the paper was published.

[@iamcute7487]

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😩

[@leonardojago7997]

It means you live forever in a dream in the universe where you remember snipits of your last life because fate takes to the same place there’s no avoiding it unless you know that already then you get your memories more fluent and some times have strange dreams tbh there’s a lot hidden about this in spirits and everything to do with magic and creation just so people know

[@isaacschmitt4803]

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.

[@TMGTurdFerguson]

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.

[@ajoeannabrown9563]

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.

[@papers744]

...nice work...thanks for sharing your idea/s about it...(all the way from a freshmen in Philippines) It would help a lot for more applications in YMIITP

[@Q_20]

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

[@mae7612]

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

[@Codermiguel]

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.

[@user-vz2wn4mq1b]

why these interesting topics are not told in school

[@samusamu1998]

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

[@Tippex_Official]

I was gonna say something like that

[@mangai3599]

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

[@yosiafirmansyah3472]

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

[@gamegirl8722]

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

[@harikalatheeswaran9206]

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 😊!

[@isyfanihdaaulia8300]

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

[@NotCommon_Foh_1795]

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

[@hungvan6771]

Still Watching it in 2019

[@gavinmann4152]

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

[@ranjankumarsahu998]

This might be one of the most enlightening videos of Teded

[@mikealdaccache8256]

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.

[@halfcolombian72]

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

[@eljah]

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

[@pakins.]

They pointed that out, in the form of the exponents of 11.

[@deepakkumarshaw8958]

no, it will work only till 4th row i.e (11)^4=14641 when it comes to the 5th row.....(11)^5=161051 but the elements of 5th row are 15101051

[@rafikhan908]

You typed 11! which represents 11 factorial

[@phuonghuynhanh9879]

@@rafikhan908 lol

[@official-obama]

@@deepakkumarshaw8958 that’s because it has more than 1 digit numbers When you do the multiplication of the powers of ten thing you get the correct answer

[@karlakrmek5613]

One of my favourite videos on internet!!

[@sheulidas7227]

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

[@YuriDokiDoki]

4:05 I learned that in math class

[@Finalounet]

Math is love, math is life

[@yp5387]

Charles C. And enemy also

[@divyaganesh1585]

Math is insanity . Math is death.

[@eunicefelix4413]

Life is difficult ._.

[@pardeepgarg2640]

.___.

[@naftalyelmescany3179]

Thank you for exist ted-ed.

[@stephenchangala6330]

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

[@yellowtinted]

Reminds me of Vihart lol

[@TheRedstoneTaco]

A vihart fan! Whats up!

[@yellowtinted]

Ey there :)

[@yovliporat8608]

triangle triangle triangle

[@TheRedstoneTaco]

3 triangles and 3 sides to a triangle 3 times three is 9 and there were two 3s so 9 to the negative 2 power equals 1. a is equal to 1 if we are talking about the numerical alphabet codex and you have a one in your name. assuming that we are talking about the number 1, 2, and 3, we could say we are starting the fibbonacci sequance and the second non one number in it is 3. three minus 1 is 2. 2 equals be in the numerical alphabet. b is the first letter in bee movie. BEE MOVIE CONFIRMED :P

[@LyricNear]

Snake, snake, snake, snake, snake, snake, snake, snake!

[@direidino8917]

In Italian we say "Triangolo di Tartaglia"

[@tommeng6522]

In mexico,, we call Avogadros number "Guacamole"

[@loremazzu408]

Che bello trovare anche qualche Italiano appassionato in matematica nei commenti

[@bravemortal]

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

[@MAGh-py9xb]

The animations are just amazing !

[@KINGKOHIRR]

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

[@johannd1100]

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

[@isaacaguilar5642]

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

[@damianskrok1731]

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

[@Hazaroth]

I love math but people who say maths is boring will likely find this video boring as well

[@VincentRubinetti]

Absolutely superb animation.

[@lion_san6493]

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

[@SKyrim190]

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

[@havefundontbeboring2822]

preach

[@azraelle6232]

It's a different way of visualizing the information. Some people find it easier to think in terms of 11's multiples, other find polynomials easier. Tomato, (2t+2o+m+a).

[@Cohnan13]

Everything in it is redundant, in the sense that you can link it all to a the definition. Now, what a particular person might believe is obvious given other patterns is what may vary depending of your understanding of the concepts

[@scaltheceiling]

oh look my triangle!:3

[@alextrebek8293]

i love math. some people see it as cold and rigid but i see it as something that shows order, thought and reason in the universe

[@Droner5]

Loved your work thank you

[@NoriMori1992]

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.

[@btdpro752]

And what...

[@hybmnzz2658]

Hahaha

[@rajeshsharmajaipur]

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

[@skylarnoel487]

Me too!!

[@Sarabheswaran7010]

👏👏👏

[@jameshanson536]

American here. I am impressed at how India has produced so many outstanding mathematicians. Ramanujan must be smiling as he looks down on that long legacy. Jai Hind!

[@rajeshsharmajaipur]

@@jameshanson536 Thank you too, your country USA is also great Nation so much contributed in education, science and technology...

[@user-gk9ui5kh5g]

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

[@zeon1966]

Amazing!!.

[@danxfabsab9331]

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

[@AshtonSnapp]

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

[@robiaus]

Brilliantly animated!

[@sallykang5475]

I needed this video!!

[@lostcat3131]

Pascal's triangle please notice me!!!

[@fighter4510]

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)