[Discrete Mathematics] Binomial Theorem and Pascal's Triangle

In this video, we look at the Binomial Theorem and how it relates to Pascal's Triangle.
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Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.

[@BiranchiNarayanNayak]

Awesome tutorial..... i just loved the Binomial theorem using pascal triangle....
I had never learnt in my high school course...

[@hannahweiss8558]

hope this saves my grade thank u so much

[@rafyyd461]

thank you so much! binomial theorem using pascal's triangle blew my mind

[@cc-to2jn]

i feel too powerful after this

[@manansood8498]

Awesome. it feels like a power

[@jonathanhernandez3628]

man haha same thats how i felt

[@LaneSurface]

Grimaldi gives an interesting proof of the binomial theorem. He says that in the expansion of (x+y)^n=(x+y)(x+y)...(x+y), then for the term x^k*y^(n-k), there are C(n,k) ways to select k x's and C(n,n-k) ways to select n-k y's. Since C(n,k)=C(n,n-k), that is the number of repeated x^k*y^(n-k) terms (or, in other words, its coefficient). Thus we can express the k-th term as C(n,k)*x^k*y^(n-k). You can also do this by induction, but it's not as simple as the combinatorial proof.

[@dianampm99]

This is a really good video! I wasnt hoping to learn all the things I needed in one video tbh, and ended up closing all the other tabs i thoguth id had to watch. thanks!!

[@skjbanks]

When you first go over Pascal's triangle you begin the numbering with a zero, then you start with a one with your first example

[@JosephSmith-n1w]

Interesting thing about the values from Pascals Triangle and the powers is, at least in the example given at with (x+y), each leading value is like taking the derivative of x and integral of y at the same time. When we take the derivative we multiple the lead value and subtract 1 from the power, and when we take an integral we add 1 to the power and divide the leading value by that new power number.
Example:
1x^6 (derivative of x^6 is 6x^5, integral of y^0 is 1/1*Y^1) so next in line is 6/1 x^5y^1,
then do it again for 6x^5y^1, derivative of x^5 is 5x^4, integral of y^1 is 1/2*y^2, so 6 * 5 *1/2 is 15 -> 15x^4y^2
15x^4y^2, derivative of x^4 -> 4x^3, integral of y^2 -> 1/3*y^3, 15 * 4 * 1/3 = 20 -> 20x^3y^3
20x^3Y^3 -> derivative of x^3 -> 3x^2, integral of y^3 -> 1/4*y^4, 20 * 3 * 1/4 = 15 -> 15x^2y^4
15x^2y^4 -> derivative of x^2 -> 2x, integral of y^4 -> 1/5 * y^5 -> 15 *2*1/5 =6 -> 6xy^5
6xy^5 -> derivative of x -> 1, integral of y^5 -> 1/6 * y^6, 6*1*1/6 = 1 -> 1y^6
Add them up and you get x^6 + 6x^5y + 15x^4y^2 + 20x^3y^3 + 15x^2y^4 + 6xy^5 + y^6
Don't think there is some correlation or helpful usage of this, just something that was interesting that I noticed.

[@Fati-y6e]

Mind power 💪🙌

[@FlakkoBC]

ur a legend Trev!

[@rkcst6503]

@7:24
Where did the (-b)^1 go???
You just treated it as if it was equal to -1... Is that correct? I must be missing something. Is any negative number raised to 1 going to be -1?

[@rkcst6503]

Wait, and why the b^2 set to just b???? I am so lost!

[@rkcst6503]

omg..... it was just a mistake lol. You fixed it like 30 seconds after and I paused before I got that far cause I was so lost lol. Thanks anyways. And thanks for the videos!!!

[@II_xD_II]

@@rkcst6503 thx for this comment I was at the 2nd part of your progress lol just searching for the similar comment
got exact comment

[@joflix3846]

God Bless You!!!!

[@II_xD_II]

that's way too true

[@drainedgaia]

4:50
I perceive it as plugging in each term of the equation as whole, when dealing with the binomial theorem.
14:21
You almost had me right there :)

[@standowner6979]

You are amazing! Thank you!!!

[@zakariabenjelloun5338]

nice video i love it thank you you explain very well

[@zakariabenjelloun5338]

yes i agree with you my friend he helped me a lot

[@zakariabenjelloun5338]

i agree with you two me too

[@itsnour1017]

aprecio eso gracias

[@asifimtiaz9310]

Beautiful

[@jonathanmalmberg787]

There's something I don't quite understand at ca 3:40. Why do you get 0! when 2 chooses 2? Don't we use the formula
n!/k!(n-k)! ? And why isn't 4 chooses 4 0! ?

[@Trevtutor]

+Jonathan malmberg 0! = 1. (2C2)=2!/(2!0!)=1=0!.

[@jonathanmalmberg787]

I see! Thank you very much. These lessons are super helpful!

[@HaiderMatrix]

thank you so much

[@abrarmahi]

casually explained: discrete math

[@zYaaaaan]

hey....
How do you get 1(16aPOWER4)?

[@Trevtutor]

+zYaaaaan (4 choose 0) is 1, (2a)^4 is 16a^4, and (-b)^0 is 1. Multiply those together.

[@aboogiewithdahoodie]

13:03 Starting with 0, still on coding/programming 😂

[@sunnyzhu5791]

i can understand 70%, who understand 100%

[@Forkythechanter]

You sounds like blue one brown

[@II_xD_II]

omfg yes sir he do lol

[@standowner6979]

I can't tell!

[@leozhang1340]

3:03 CHU TU ziro ;P

[@joecremeno1078]

your theorem is reversed, you did not make me happy.