Matt Explains: Binomial Coefficients [featuring: choose function, pascal's triangle]

This is a Matt Explains video. Still very much a work in progress. All feedback hugely appreciated.

Sorry about the strobing camera focus on the whiteboard! I seem to be in focus this time, but at the cost of my camera freaking-out over the board. I might need a new camera. Or a camera person. Whichever is cheaper.

Read all about the Binomial Theorem at MathWorld:

[tone]

Best guess at the Pascal's Triangle:

This goes back to what you were doing at the beginning, expressing each binomial expansion as the previous one multiplied by another (A+B). In that system, each of A and B gets multiplied by each of the previous parts, and the sum of all those results gives us our next expansion.

So let's look at the third expansion: A^3+3(A^2)B+3A(B^2)+B^3. To find our fourth, each of those gets separately multiplied by an A and a B. To simplify, let's look at that with just the first two terms: (A^3+3(A^2)B)*(A+B).

If we multiply A^3 by A, we get A^3. If we multiply it by B, we get (A^3)B. If we multiply 3(A^2)B by A, we get 3(A^3)B. If we multiply it by B, we get 3(A^2)(B^2). Now the question is, how many (A^3)B results did we get, total? Well, we got one for every A^3 in the previous expansion (1 total), plus one for every (A^2)B in the previous expansion (3 total). No other variable segment can produce (A^3)B, so our result is the sum of those two terms. Since each variable segment can only become one of two different variable segments in the next expansion, this should happen with any other pair of terms as well. (This one also gave us a bonus term because the A^4 term can only be approached by one segment, but that won't happen with more central segments.)

That took a lot of work to figure out. I should do this stuff more often...

[NALGames]

I swear to god that if I'd had you as an AS Maths teacher instead of the disinterested person I did get, I wouldn't have miserably failed it haha. Another superb video, entertaining and well laid out :) Thanks, and I hope to keep seeing em coming!

[TrimutiusToo]

So nice of you to say that my comment was your favourite =)
Regarding Pascal triangle it is pretty easy to see why it works
In binomial coefficient we have for example
1 3 3 1
Then when we multiply by (a + b) if we consider the result it will be
1 3 3 1 0
+
0 1 3 3 1
=
1 4 6 4 1
Because powers will be shifted by a or b, so we sum up previous coefficient with itself but shifted by 1 position. And pascal triangle does exactly that, you sum up (k, n) position with (k, n+1) position in order to get the position (k+1, n+1) (where k is the number of row in Pascal triangle), and positions of (k, 0) and (k, k) are always 1 (which even works for 0's row, because 1 = 1)


P.S. Binomial Coefficient is one of my favourite subjects though, I was so interested in this concept that I actually asked my dad (who has PhD in Physics) to explain it to me when I was in 3rd Grade... Though I asked him to explain the Choose Function back then (even though I didn't know the word "function" back then), only later on I found out about Binomial Coefficients and Pascal Triangle. And I personally prefer "Binomial Coefficient" or "Newton's Binomial Series" names.

[PaoloSilverInzaghi]

Pascal's triangle answer: the number of paths you can take from the top to a number. i.e. you can take 6 different paths to get from the 1 at the top to that number 6.

What a fun way to wake up on this post finals day. :D Thank you very much, Matt!!

[henk]

I noticed the Pascal's triangle thing when you wrote the numbers down, but I couldn't figure out why. I'm really looking forward to you explaining why :)

[cartmanthesecond]

It's clear from the definition of the choose function that for any natural number n,

C(n, 0) = C(n, n) = 1.

It is relatively straightforward to show - using the definition of the choose function and some algebra - that for any natural number m such that 0<m<n:

C(n, m)=C(n-1, m-1) + C(n-1, m).

By showing this, and noticing that the first two rows of Pascal's triangle list the choose functions for 0 and 1, it follows from the definition of Pascal's Triangle that each subsequent row will list the binomial coefficients for the subsequent powers.

I have no idea what the intuition for this is, though.

[alejandronq]

At this point i have to:
Zeroth: wander what the hell will happen if you make a triangle with trinomial coefficients.
1/2th: sugest it has to do with "Pascal's tetrahedron"
First: admit you've blown my mind
Second: thank you this awesome video
Third: point out that at the time I'm writing this video has 234 likes and 2 dislikes

[headrockbeats]

Most of video: *"Huh, pretty cool that it works like that. Makes sense."*
9:45: *MIND->BLOWN.*

[Crazyfingers]

Thank you so much for explaining this concepts and connection. I wish every teacher in the U.K teaches maths like you did in this video .

[ffggddss]

Another truly *standup video!* Splendid! You never sat down once!
I was breathlessly waiting for you to show the direct connection between Choose and Binomial by showing the Choose function being used to 'assemble' each term in the expansion.
And son-of-a-gun if you didn't do exactly that! Kudos, Matt, and thanks!

Fred

[tippybooch]

Love these videos so far, it's like having extra Numberphile videos to watch! I just wanted to ask if I may, do you have an intuitive understanding of maths in any way or is it something you've had to work extra hard on? I ask because while I love maths I really struggle with it, which is why I love videos like this. Makes it all that bit easier to grasp.

[Theraot]

When you are at 121 it means that you have 1 way to have aa 2 ways to have ab and 1 way to have bb. When you start the 1331 the first 1 is the number of ways to get aaa, which is the 1 way to get aa with an extra a. The first 3 is the number ways to get aab, and that is the 1 way to get aa with an extra b plus the 2 ways to get ab with an extra a. And that is - by example - why you can get the coefficients by adding pairs of the coefficients for the previous power.

[tommysadler]

Imagine if this guy was my Maths Teacher. So good!!!! :) :)

[joykeenan]

Matt you're a legend. I've been struggling with this for the past month and I finally understand.

[RexGalilae]

Apart from the fact that I have this geniunely funny guy explaining Math to me, I'm also satisfied that he cares to explain "why". This is something noone does.

[seb]

I had a few things to do today... but instead I spent most of the day watching your videos. Those are truly awesome :)

[KalikiDoom]

Awesome video matt! - thank you for sticking with it,
and don't forget to do a standupmaths about pascal's triangle.

[jamestwosheep]

Aw yiss. I used to always expand binomials with Pascal's Triangle. It used to saved me lots of time in exams. Ah, fond memories.

[Rahul.G.Paikaray]

I'm from indian you are great explanation of this topic
Keep it up ALWAYS
You know ours mathematicians. Like RAMANUJAN. ARYABHATTA. SAKUNTALA DEVI
All are genius
You also like that
Thanks to give your time to read my comment
I salute you

[sagy]

I'm a college student (year 13 UK) and I just want to say I love your videos and that I just did this in class Monday
I found this by chance and it has really helped thanks for the great videos