Binomial expansion & combinatorics

what I want to do in this video is hopefully give more intuition as to why the binomial theorem or the binomial formula involves combinatorics so let's just think about what this expansion would be so I'm just using a particular example that's pretty simple X plus y to the third power which is X plus y times X plus y times X plus y so we already know what the terms look like we could pick an X from each of these so if we pick exactly an X from each of these that's going to be our X to the third term and it's going to be and we essentially picked no wise n because we picked an X from each of these when we multiply it out to get this term and how many your ways are there to construct that how many ways are there if we're picking from three things so we're taking we're picking from three things how many ways can we choose exactly 0 wise of not picking a Y well there's only one way to do that there's only one possible way of doing that by taking and by not picking a Y from each of them or you could say by picking an X from each of them so the coefficient right over here is 1 X to the third is only one way when you take this product out only one of the terms initially is going to have an X to the third in it now what about the next term so plus so what if we're now we're going to pick from three buckets but we're going to we want to pick one Y so another way of thinking about it you have three people or we have three people this person this person and this person and one of them you're going to choose to be on to go to I guess you know be your friend that's another way of saying which of these are you going to pick the Y from so this is this is the exact same thing as the combinatorics problem from three things from three things you are choosing one of them to be Y and of course 3 choose 1 this is equal to 3 and so if you're if you're picking one Y that means you're picking two x's so and you're taking the product of everything so it's x squared times y to the 1 and we could keep doing that logic so + so now we're going to think about the situation where there we are picking two wise out of three things so it's X to the first y squared so we're starting with three things and we're going to pick two to two of these Y so it could be this Y and this Y to take the prot to construct the product Y squared so it could be y times y times that X or it could be it could be Y times y times that X or it could be Y times X times y so this is 3 choose 2 so out of 3 things what are the different ways what are the combinations of picking two things if you have three friends how many combinations can you put two of them in your two-seater car where you don't care about who sits in which seed you're just think about how many different combinations can you pick from that and once again this evaluates to three and then finally how many different ways how many different ways from a set of three things from three different things so from each of these expressions how many ways can you pick exactly three wise well there's only one way to do it you pick this Y this Y and that Y and so if you're picking three wise that means you picked zero X's and you have picked three Y's so that's why we're dealing with combinatorics here three choose three from from three expressions your three binomials right over here that you are taking the product from you're choosing you need to choose three Y's so you have to pick the second term in all of them there's only one way to do that here you want to pick the second term the y term in two of them exactly two of them here you want to pick the y term in exactly one of them there's one way to do this three ways to do this sorry there's three ways to do this three ways to do this one way to do that one way to do that so this is one this is three when you evaluate when you apply the formula for three choose or you know n choose K three and this is one