Binomial theorem
Binomial Theorem (part 2) Binomial Theorem and Pascal's Triangle
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- In the last video we saw that if you wanted to take a plus b
- to the nth power, and if n is larger than, really, 2-- but
- really, especially larger than 3-- it is very tedious to
- multiply it out, essentially using the distributive
- property, or doing polynomial multiplication, or FOIL, or
- however you learned it.
- It is extremely, extremely tedious.
- And then we learned that the binomial theorem, which said
- this, that that is equal to the sum from k is equal to 0,
- to n of n choose k-- right?
- Where that was what we learned in combinatorics as the
- binomial coefficient.
- And that's why it's called a binomial coefficient, because
- it's actually the coefficient of the binomial theorem.-- Of
- x to the n minus k-- oh, sorry, I keep writing x.
- Let me undo that.
- Edit, undo.
- Edit, undo.
- Oh, it's taking too long.
- Oh, let me just-- oh no, that's not what I wanted to do.
- Let me erase it.
- OK.
- I keep writing x.
- It could be an x, but then this would have to be
- an x here, as well.
- Maybe I should do that.
- --Of a to the n minus k, times b to the k.
- So each term-- you know, the n stays constant-- but each term,
- you start at k equals 0 and you keep incrementing up.
- And we did an example to solve a plus b to the fourth power
- in the previous video.
- And as you saw, that was tedious, but less tedious than
- actually multiplying it out.
- And if you get really fast at computing n choose k
- for different ns and k, it could be reasonably fast.
- So what I want to do is, I'm going to show you a slightly
- faster method than what we just did.
- Kind of a faster way to compute the binomial coefficients.
- And then after that, I'm going to show you a super fast way
- that, short of memorizing the coefficients-- which I actually
- know some people who've done that-- is a pretty amazing way
- to essentially multiply out any binomial.
- So what's my pseudo fast way of doing it?
- Well, I hinted in the last presentation that those
- coefficients were actually terms of a Pascal triangle.
- So what's a Pascal triangle?
- So if we start off with a 1 and then you just go-- actually,
- let me do it-- well, yeah, let me do it right here.
- And then, actually let me start with two 1s.
- And what you do is, you take the sum of both of these.
- So that's a 2.
- And then you bring down a 1, to the left and
- the right hand side.
- And notice, these are the coefficients of
- a plus b squared.
- And these are the coefficients of a plus b.
- You could say a plus b to the one.
- 1a plus 1b.
- This is a squared-- so you could rewrite, a plus b
- squared. is 1a squared plus 2ab plus 1b squared.
- So these are the coefficients of a plus b squared.
- Let me arbitrarily switch colors.
- And so, 1 plus 2 is 3.
- 2 plus 1 is 3.
- Bring down the 1.
- Bring down the 1.
- And now we have the coefficients for a
- plus b to the third.
- Which we computed in the-- that was the very
- first thing we did.
- We actually multiplied it out.
- And we just know the pattern.
- The first coefficient is 1.
- So it's 1a to the third, b to the zero-- so we don't have
- to write the b-- plus 3.
- We just decrement this exponent 1.
- 3a squared b plus 3ab squared, and then plus 1a to the zero--
- which is just 1-- b cubed.
- So that was pretty fast.
- And we can keep going down the Pascal triangle.
- So let's do the next one.
- So we can bring down a 1.
- 1 plus 3 is 4.
- 3 plus 3 is 6.
- And this is neat.
- I mean, just very simply, you can actually generate binomial
- coefficients without having to compute them.
- Very simple, I guess you could call it an algorithm.
- Or drawing.
- And it's symmetric, just as you would expect it, right?
- Because you could easily switch b and a.
- a plus b is the same thing as b plus a, so you
- should essentially get the same answer.
- And so, we just-- very quickly we figured out the binomial
- coefficients for a plus b to the fourth.
- Which was a lot faster than we did in the last example.
- a plus b to the fourth.
- So then we-- I think you get the point.
- But, so it's 1-- let me write in a different color-- 1a
- to the fourth, b to the zero, plus 4a cubed b
- squared-- b to the one.
- Plus 6a squared b squared.
- Which makes sense that, you know, this is a middle number
- and they both-- and a and b have the same exponent
- at this point.
- And then plus 4a-- we decrement it-- b cubed.
- Plus b to the fourth.
- 1b to the fourth, right? a to the zero, so that's what
- we didn't write there.
- So 1b to the fourth.
- And that was very fast compared to what we had to do at
- the end of the last video.
- We could just keep going.
- You know, for 5.
- So 1 plus 4 is five.
- 4 plus 6 is 10.
- 6 plus 4 is 10.
- 4 plus 1 is 5.
- Bring down the 1.
- So these are the coefficients for the expansion of a
- plus b to the fifth power.
- And so this is a reasonably fast way of doing it, although
- it can get-- one, it will take a lot of space.
- And it can work reasonably well, you know, for up to a
- power of eight or nine or ten.
- Even then it starts to get pretty big and cumbersome.
- But you know, for powers up to seven or eight or
- nine, you could do this.
- You could draw it out really fast and do this, and it's
- probably faster than actually computing each of the
- binomial coefficients.
- Although you might be pretty fast at computing n choose k,
- in which you don't have to do this.
- So with that out of the way, let me show you an even faster
- way of doing it, short of memorizing it.
- And this will allow you to really calculate a plus b
- to the nth-- you know, to the twentieth power--
- almost in your head.
- Depending on how good you are at arithmetic in your head.
- So here is the trick.
- And I encourage you to experiment for why it
- works, but it does work.
- And I mean it's not even a trick.
- It's just-- and this Pascal's triangle isn't even a trick.
- Pascal's triangle is just an alternative way to generate
- binomial coefficients, and what I'm about to show you is just
- another way of essentially generating the binomial
- coefficients.
- Although it's probably a faster way to compute them.
- And it's a good project for you to think about why this works.
- So I'm just going to start with a very concrete example.
- Instead of a plus b, let me just do x plus y.
- Just because you might see the binomial theorem
- written that way.
- So let's say x plus y to the tenth power.
- This would take me all day if I was to actually
- multiply it out.
- It would take me probably 20 to 30 minutes, without
- making careless mistakes, to actually figure out all the
- binomial coefficients.
- Maybe not that long, but it would take me a while.
- And to draw Pascal's triangle would fill up a whole page,
- and I'd still probably make a careless mistake.
- So how can I do this?
- So what you do is-- so one thing you know.
- This is going to have 11 terms, right?
- Because you're going to start with x to the
- tenth, y to the zero.
- And you're going to go all the way to y to the tenth.
- So if you start at 0 and you go to 10, that's 11 terms.
- So it has 11 terms.
- What I want you to do is just write down the
- first, just the numbers.
- You know, you can almost count the terms.
- You don't have to go all the way to 11, and I'll show you.
- But actually, let's write all the way to 11.
- So 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.
- Just squeezed it in.
- And you'll see, you don't actually have to go
- all the way to 11.
- You could probably just stop at 6.
- So here's the trick.
- We know that the first term is going to be x
- to the tenth, right?
- We know that x to the tenth.
- Actually, we know that it's going to be x to the tenth.
- The second term is going to be x to the ninth.
- Then it's going to be x to the eighth.
- It's going to be x to the seventh.
- A little tedious. x to the sixth.
- x to the fifth.
- x to the fourth.
- x to the third.
- x squared.
- x.
- And then there's going to be x to the zero, or just one.
- Let me just do the y's.
- So this was x to the tenth.
- That's not bright enough, this color.
- So this is y to the zero.
- So we don't have to write it there.
- But then we have a y.
- y to the first.
- y squared.
- y to the third.
- y to the fourth.
- y to the fifth.
- Which makes sense, this is the middle term.
- y to the sixth.
- y to the seventh.
- y to the eighth.
- y to the ninth.
- I don't want you to get confused, each of these
- is a separate term.
- I don't want you to think I'm multiplying them all.
- And then, we just have to figure out the coefficients
- on each of these terms.
- Those are divider lines I attempted to draw.
- I wasn't trying to confuse you more.
- I just wanted to-- because they seemed to be running together,
- each of the terms that I'm writing.
- But I think you know what I'm doing.
- So now we have to figure out the coefficients.
- And then this is the neat part.
- So we know that the coefficient on the first term-- let me draw
- a dividing line here and here-- the coefficient on the first
- term is always 1, right?
- So, the coefficient is 1.
- So the coefficient on the second term is going to be the
- exponent on the first term times its coefficient-- so 10
- times 1-- divided by the term that it is.
- So it's going to be 10 times 1, divided by 1.
- So it's going to be 10.
- The third term's coefficient is going to be the
- exponent on the x, right?
- So it's 9 times its coefficient-- which is 10-- so
- it's going to be 9 times 10-- divided by the term it is.
- So it's going to be 9 times its coefficient, 10, divided by 2.
- So what's 9 times 10?
- That's forty-- it's 90 divided by 2 which is 45.
- And you keep going.
- The fourth term is going to be the third term's exponent-- so
- it's going to be 8 times-- let me write this down in a
- different color-- it's going to be 8 times its coefficient,
- times 45 divided by which term it is.
- So it's the third term.
- Divided by 3.
- Well, that's just 8 times 15.
- And we'll see, that's 80 plus 40.
- So that's equal to 120.
- So that is the fourth term.
- And so then let me just draw these dividers.
- I know it's getting a little complicated.
- And I'm writing it all out like this, but if you practice
- this enough, you can actually just write it straight out.
- And so the fifth term.
- What is the fifth term?
- Well, you take the exponent on the x.
- So 7 times the fourth term's coefficient--
- times 120-- divided by 4.
- Right?
- Divided by the previous term, by 4.
- Well, that's just 7 times 30, which equals 210.
- That's the fifth coefficient.
- What's the sixth coefficient?
- Well, it's 6 times-- you know the exponent on
- the x-- times 210.
- Times its coefficient-- times the fifth term's coefficient--
- divided by 5, for the fifth term.
- Well, 5 goes into 210 how many times?
- 42 times, right?
- So it's 6 times 42-- that's 240 plus 12.
- That's 252.
- And then once you're at the middle point-- the sixth term
- is the middle term-- you'll see that, you know, you start
- going back the other way.
- And we learned one from the Pascal's triangle, or even the
- definition of the binomial theorem, that the
- coefficients are symmetric.
- So we know that the next one is going to be the same-- this was
- the middle one, right?-- so we know the next one is
- going to be 210.
- And you could calculate it using the same system.
- This is just a quick way of doing it.
- This one's going to be 120.
- This one's going to be 45.
- And this one is going to be the tenth coefficient--
- this is going to be 10.
- And then of course the last coefficient is just 1.
- 1y to the tenth power.
- So if I were to write this out, the answer is-- and if you
- practice this, you'll find that you can do quite fast-- it's x
- to the tenth, plus 10x to the ninth y, plus 45x to the eighth
- y squared, plus 120x to the seventh y to the third, plus
- 210x to the sixth y to the fourth, plus 252-- already at
- the middle term-- x to the fifth y to the fifth, plus 210x
- to the fourth y to the sixth-- I'm running out of space.
- But you can you can hopefully extrapolate what I'm doing,
- and it makes sense to you.
- And hopefully you have an appreciation that if you
- actually had to multiply out x plus y to the tenth, it would
- have taken you all day.
- Maybe I'll do one more video with a smaller example to show
- you that it's a little less complicated when you do, say,
- x plus y to the sixth power.
- See you soon.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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