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Current time:0:00Total duration:8:20

Pascal's triangle and binomial expansion

CCSS.Math:

Video transcript

in the previous video we were able to apply the binomial theorem in order to figure out what a plus B to the fourth pirate is in order to expand this out and we did it and it was a little bit tedious but hopefully you appreciate it it would have been useful if we did even a higher power a plus B to the seventh power a plus B to the eighth power but what I want to do in this video is show you that there's another way of thinking about it and this would be using Pascal's triangle and if we have time we'll also think about why these two ideas are so closely related so instead of doing a plus B to the fourth using this traditional binomial theorem I guess you could say formula right over here I'm going to calculate it using Pascal's triangle and some of the patterns that we know about the expansion so once again let me write down what we're trying to calculate we're trying to calculate a plus B so a plus B a plus B to the fourth power to the fourth power I'll just do this in a different color to the fourth power so what I'm going to do is set up Pascal's triangle so Pascal's triangle so we'll start with a one at the top and one way to think about it is if it's a triangle where if if you started up here at each level you're really counting the different ways that you can get to the different nodes so one and so I'm going to set up a triangle so if I start here there's only one way I can get here there's only one way that I could get there but now this third level there are if I were to say how many ways can I get here well one way to get here one way to get here so there's two ways to get here one way to get there one way to get there the only way I can get there is like that the only way I could get there is like that but the way I could get here I could go like this or I could go like this and then we could add a fourth level where let's see if I have there's only one way to go there but there's three ways to go here one plus two how are there three ways you could go like this you could go like this or you could go like that same exact logic there's three ways to get to this point and then there's only one way to get to that point over there and so let's add a fifth level because this is actually what we care about when we think about something to the fourth power this is essentially zeroth power binomial - zeroth power first power second power third power so let's go to the fourth power a fourth power so how many ways are there to get here but I just have to go all the way straight down along this left side to get here so it's only one way there's four ways to get here I can go like that I could go like that I could go like that and I can go like that there's six ways to go here three ways to get to this place three ways to get to this place so six ways to get to that and if you have the time you could figure that out there's three plus one four ways to get here and then there's one way to get there and now I'm claiming that these are the coefficients when I'm taking something to the so this is this is if I'm taking something to the zeroth power this is if I'm taking it or not something a binomial to the first power to the second power and you might obviously by normal to the first power of the coefficients on a and B are just one and one but when you square it it would be a squared plus 2 a B plus B squared if you take the third power these are the coefficients third power and the fourth power these are the coefficients so let's write them down the coefficients I'm claiming are going to be one four six four and one and how do I know what the powers of a and B are going to be well I start a I start this first term at the highest power at a to the fourth and then I go down from there a to the fourth a to the third a squared a to the first and I can guess I could write a to the zero a to the zero which of course is just one and then for the second term I started the lowest power at zero and then B to the first B squared B the third power and then B to the fourth and then I just add those terms together and I just add those terms together and there you have it I have just figured out the expansion of a plus B to the fourth power a plus B to the fourth power is exactly what I just wrote down this term right over a to the fourth that's what this term is one a to the fourth B to zero that's just a to the fourth this term right over here is equivalent to this term right over there and so I guess you you you see that this gave me an equivalent result now an interesting question is why did this work why did this work and I encourage you to pause this video and think about it on your own well to realize why it works let's go let's just go to these first levels right over here if I just were to take a plus B to the second power a plus B a plus B a plus B to the second power this is going to be we've already seen it this is going to be a plus B times a plus B so let me just write that down a plus B times a plus B so we have an A and a we have a B and a B and we're gonna add these together and then when you multiply it you have so this is going to be equal to a times a so you get a squared and that's the only way that's the only way to get an a squared term there's only one way of getting an a squared term then you're going to have plus a times B so let me write this so plus plus this way plus a times B and then you're going to have Plus this B times that a so that's going to be another a times B a times B plus plus B times B which is B squared now this is interesting right over here how many ways can you get an a squared term well there's only one way you're multiplying this a times that a there's only one way of one way of getting there now how many ways are there of getting the B squared term how many ways of there getting the B squared term was only one way multiply this times this B there's only one way of getting that but how many ways are there of getting the a B term the eight of the first B to the first term well there's two ways you can multiply this a times that B or this B times that a there are just hit the point home there are two ways two ways of getting an a B term and so when you take the sum of these two you are left with a squared plus 2 plus 2 times a B 2 times a B plus B squared notice the exact same coefficients 1 2 1 1 2 1 why is that like that well there was only one way to get an a squared there's two ways to get an a B and there's only one way to get a B squared if you send it to the third power you would say okay there's only one way to get to a to the third power you just multiply the first ace all together and there is only one way to get to B to the third power but there's three ways to get 2a squared B and you could multiply it out and we did it we did it all the way back over here there's three ways to get a squared B we saw that right over there and there are three ways to get a B squared three ways to get a B squared and if you sum this up you have the expansion of a plus B to the third power so hopefully you found that interesting