Binomial theorem
Binomial Theorem (part 1) Introduction to raising (a+b)^n
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- We'll now learn an application of combinations that you
- probably won't find intuitive at first.
- But the more you think about it, it will make a lot of sense
- and hopefully it'll make you appreciate, once again, the
- beauty of mathematics.
- And then you'll also know why-- when we say n choose k in
- combinations --why that's also called a binomial coefficient.
- Because we are going to cover the binomial theorem.
- So before I give you the binomial theorem, let's
- understand the motivation for why it's even interesting.
- So let me erase that.
- Invert colors.
- So if we just had to multiply-- I don't know.
- Well let's just take different powers of a binomial.
- A binomial is just a polynomial with two terms, right?
- So a plus b-- Well a plus b to the 0, that's
- equal to 1, right?
- Anything to the 0 is equal to 1.
- a plus b to the 1, well that equals a plus b.
- a plus b squared, that equals-- and if you don't have a lot of
- practice doing this you might be tempted to say a
- squared plus b squared.
- But you should correct yourself quickly and slap yourself on
- the wrist or the brain or someplace if you did that.
- Because that's a plus b times a plus b.
- And then we could use the distributive property, or,
- if you learned it this way in Algebra I, you could
- use the FOIL property.
- That equals a times a plus b, right?
- Plus b times a plus b.
- Which is equal to a squared plus ab plus ba plus b squared.
- Which is equal to a squared plus 2ab plus b squared.
- That should be a bit of review for you.
- And now it gets a little bit more interesting.
- What is-- Let me circle that just so we remember it.
- That's a plus b squared.
- What's a plus b to the third?
- And now this is starting to get complicated.
- This is equal to a plus b times a plus b times a plus b.
- Or another way to view it, it's a plus b squared
- times a plus b, right?
- This is a power of three.
- So this was a plus b squared.
- So if we multiply it by a plus b, we'll get a
- plus b to the third.
- So let's do that.
- Let's multiply this times a plus b.
- So first let's multiply everything times b.
- So that's b-- let me do this in another color.
- a squared b, right?
- That's a squared times b.
- Now let's do 2ab times b.
- So plus 2ab squared, right?
- 2ab times b.
- And then plus b cubed.
- And then we have a times a.
- Well that's a cubed, right?
- None of these match that, so I'll put it in another column.
- a times 2ab.
- Well that's 2a squared b.
- I'll put that out here: 2a squared b.
- And then a times b squared.
- Well that's plus ab squared, right?
- And now we'll just add up all of the terms.
- All we do is the distributive property again, right?
- We multiplied a times all of these terms and then added that
- to b times all of these terms.
- If we add it all up-- I'll try to do it in order
- of-- Let's see.
- Let's put the a cubed first.
- And then-- Well actually we already had this thing.
- This 2a squared b, I could have written it here.
- 2a squared b, because I had an a squared b here so.
- I just rewrote the 2a squared b here.
- So we have a cubed plus 2a squared b plus a squared b.
- That's 3a squared b.
- And then 2ab squared plus ab squared.
- That's 2ab squared.
- And then plus b cubed.
- As you can see, that involved a lot just to take something
- to the third power.
- So we could-- If we had the time, we could figure out what
- a plus b to the fourth power is or what a plus b to
- the tenth power is.
- But as you could imagine, this would take you all day.
- So wouldn't it be neat if there were an easy way to calculate
- what a binomial is to an arbitrary power?
- And that's where the binomial theorem comes into play.
- And in this video I'm going to show you what the
- binomial theorem is.
- I will show you how to apply it.
- I will show you a trick or a technique that will make
- you seem like a genius.
- And then in the next video I'll hopefully give you
- some intuition for why the binomial theorem actually
- involves combinations.
- Why it involves actually the binomial coefficient at all.
- So what is the binomial theorem?
- Let me erase all of this.
- And you can confirm that the binomial theorem works for the
- ones that we've worked out, up to a plus b to the third.
- You could work out a plus b to the fourth if you
- like to punish yourself.
- Let's see.
- Clear image.
- Invert color.
- So the binomial theorem tells us that a plus b to the nth
- power is equal to-- And I know it's going to look complicated
- at first, but we'll do a couple of examples and you'll see
- it's not that intimidating.
- It equals the sum from k equals 0 to n, right?
- This n is the same thing as that n.
- Of-- each term is n choose k, right?
- We're going to keep incrementing k up from 0 to
- n-- of x to the n minus k times y to the k.
- I know that looks complicated but if we do a couple of
- concrete examples, I think it should make a reasonable
- amount of sense.
- So given-- Oh sorry.
- This is-- This isn't-- I was copying this down.
- This should be a to the n minus k and this should be b.
- What I had written down before, that would be
- x plus y to the n.
- If we have a plus b to the n, n choose k each term. a to the
- n minus k times b to the k.
- So let's apply this, a couple of concrete examples.
- We could even switch around the variable names if we want just
- to show you that they don't have to be a's and b's.
- They can be anything.
- So what is a plus b-- let's do one that we otherwise would
- have found fairly difficult-- a plus b to the fourth power.
- Well that, the binomial theorem tells us that, let's see, the
- first term is going to be-- Well what's n, first of
- all? n is 4 in this case.
- It equals-- Let me fill in all the numbers actually.
- From k equals 0 to 4 of 4 choose k, right?
- Because k is what we're incrementing.
- a to the 4 minus k.
- b to the k, right?
- I just substituted the n into the binomial
- theorem definition.
- And what does that equal?
- Well the first term is k equals 0.
- So that's 4 choose 0.
- So out of 4 things, I'm going to choose 0.
- And I'll show you in the next video why that works.
- Of a to the 4 minus k.
- Well the first term k is 0.
- So it's a to the fourth, b to the 0, right?
- So the b, that's just 1, so we can just ignore it.
- So what's the next term?
- Well, it's going to be 4 choose 1.
- And now k is 1.
- So 4 minus 1 is 3.
- a to the third.
- And k is 1 now.
- We're in the-- This is the zeroth term.
- This is the first term.
- So b to the first plus-- So as you can see, each term we go,
- the a term, the first term, whichever it is, it decrements.
- It starts up at the power n, or in the fourth power.
- And then each term, it goes down by 1.
- And then the second term, the b term, it starts
- at the zeroth power.
- So it starts at 1.
- That's why you don't see it there.
- And then each term it increments up.
- So then the next one-- So I think you see the pattern.
- It's going to be 4 choose 2, a squared b squared plus 4 choose
- 3 ab to the third plus 4 choose 4.
- Now it'll have a to the zero, so that's just
- 1, b to the fourth.
- So we're done if we just figure out what these binomial
- coefficients are.
- And that's where they come from, from the
- binomial theorem.
- But we remember how to calculate that, right?
- In general-- and hopefully you have the intuition on this.
- You shouldn't just memorize it.
- n choose k from our combinatorics is equal to n
- factorial over k factorial divided by n minus k factorial.
- So in this case, what's 4 choose 0?
- That equals-- I know it seems very time consuming right now.
- Although it's less time consuming than actually
- multiplying it out.
- But I'll show you trick in a second that will amaze you.
- So this is equal to 4 factorial over 0 factorial times 4
- factorial, right-- 4 minus 0 is 4 --a to the fourth plus 4
- factorial over 1 factorial times 3 factorial, right?
- 4 minus 1 is 3 factorial.
- a to the third b plus-- I know this is getting a little
- tedious, but I think it's good to completely work through
- one entire problem --plus 4 choose 2.
- That's 4 factorial over 2 factorial times 2
- factorial, right?
- 4 minus 2 is 2.
- a squared b squared plus 4 choose 3.
- That's 4 factorial over 3 factorial.
- 4 minus 3 is 1 factorial.
- ab cubed.
- And then 4 choose 4.
- That's plus 4 factorial over 4 factorial times 0
- factorial b to the fourth.
- And notice: this coefficient is the same as that coefficient.
- This coeffieicnt is the same as this coefficient and then
- this one's in the middle.
- So let's evaluate them.
- And I'll switch colors.
- So 0 factorial, in case you don't know it, it's
- actually defined to be 1.
- Which is a little bit non-intuitive because 1
- factorial is also 1.
- But that's just something you should know.
- So 4 factorial divided by 0 factorial times 4 factorial.
- This is actually equal to 1.
- So the first term is just a to the fourth plus 4 factorial--
- it's 4 times 3 times 2 times 1 --divided by 3 times 2 times 1.
- So that equals 4.
- 4a cubed b plus 4 factorial.
- That's 4 times 3 times 2 times 1.
- So that's 24, right?
- Over-- what's 2 factorial?
- That's just 2.
- So 2 times 2 is 4.
- So 24 divided by 4 is 6.
- So 6a squared b squared plus well 4-- This term is the
- same as this term, right?
- With just the 1 factorial and the 3 factorial
- got switched around.
- And you might want to think about that for a few
- seconds as to why that is.
- It should make a little sense to you.
- But that is-- So it's going to be 4ab cubed.
- And it makes sense, right?
- Because this could have just been b plus a.
- A plus b and b plus a are the same thing so it makes sense
- that there's a symmetry, right?
- That we have 4ab cubed and we also have 4a cubed b.
- Ignore me if you find that confusing.
- If you find it enlightening, all the better.
- And then the last term.
- 4 factorial.
- This term is the same thing as this term.
- And we've already figured out that equals 1.
- So plus b to the fourth.
- So I had a little symmetry.
- The coefficients are 1, 4, 6, 4, 1.
- And I'll show you in a future video that these are actually
- the terms of a Pascal Triangle, which is another avenue
- to go in mathematics.
- But anyway, this was an application of the
- binomial theorem.
- And I realize I've taken 12 minutes so far.
- So I will do more examples in the next video.
- 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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