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.

100 thoughts on “The mathematical secrets of Pascal’s triangle – Wajdi Mohamed Ratemi

  1. Easy I solved this alone in 30 seconds
    Easily is
    First row 11^0=1
    Second 11^2=121
    So on

  2. Things they have missed…
    Fibonacci sequence and the use in Binomial Theorem.

  3. for the combinatorics part you said were looking at the 3rd number so for every row you look at the 3rd number or do you look at the highest number in that row

  4. I think Chinese make this person up. I grow up from china, I have never know something like these. I am shame for china. they all way's make thing's up. it is ok to have something they had never study. we don't need to make up a name for the area. Before 1800, china have never learn to use some people's name for certain area. not a single one. now , suddenly there is Yang Hui jump out. Shame of such action. if you are study overboard and work with other countries scholar.Please amid that china don't have such things. as everybody know, even though, Chinese good at math, in history China didn't study math. The only thing they study is abacus.

  5. 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

  6. You could say 2+1 or you could just create a table, multiply xy on the exponential rate your glucose is at in the moment and divide it by the tipe of angles a triangle of your choice has and obtain the 4th line which equals to 3, or better said 2+1

  7. would be more accurate and consistent if you had the origin dates from all known civilization who used this triangle. thanks

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

  9. Sierpinski’s Triangle is like a triforce, but every triangle was cut like a triforce as well.

  10. so India, Iran, and China to discover it and of course Europe, as usual, to take over!!

  11. Also if you take nth row and all coefficients are multiple of n, n is a prime 🙂

  12. 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

  13. 3blue1brown made a video about how possible actions in the tower (of hanoi?) Game with stacked disks correspond with skirpinscis triangle when the optimal solution is drawn as a path, an interesting fractal curve.

  14. Before I watched the rest of the video, I tried to see the pattern and I was sort of correct. A triangle of ones, and then you start from the second row on the left side. You add the left one on the second row to the number on its left. The sum goes in between them, then you do the right side. Then you repeat the process.

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

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

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

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

    Can i pass matric first

  19. 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

  20. 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)

  21. ladies and gentleman, this video here is proof that illuminati invented maths.

  22. 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.

  23. you can find more secrets of pascal triangle at this blog post
    https://www.themaths.in/2019/05/pascals-triangle-hidden-secrets-and.html

  24. 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? 🙂

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

  26. 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 😊!

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

  28. 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.

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

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

  31. 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

  32. 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

  33. Mean while in Indian Classrooms..
    Find the 7th from the last co-efficient of x in the expansion (x+y) ^2019.
    In YouTube : This.

  34. 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

  35. 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

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

  37. I was amazed by how much can be done by a single triangle😲
    For which we learn 20 formulas😤😢

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

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