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# Expanding binomials w/o Pascal's triangle

CCSS.Math:

## Video transcript

what I want to show you in this video is what could be described as I guess a trick for finding binomial expansions especially binomial expansions where the exponent is fairly large but what I want you to do after this video is think about how this connects to the binomial theorem and how it connects to Pascal's triangle so now let me show you the trick so I'm going to take X plus y to the seventh power that's going to have eight terms how do I know that well X plus y to the first power has two terms it's a binomial X plus y squared has three terms X plus y to the third power has four terms so this is going to have eight terms so let me just create little buckets for each of the terms so this is the bucket these aren't the coefficients these are just the buckets first term second term third term fourth term fifth term sixth term seventh term and eighth term now let's write out the actual let's write out the actual X's and Y so the first term we're going to start with X to the seventh power then each consent each each term after that our degree or our power on the X goes down by one so you go X to the sixth X to the fifth X to the fourth X to the third x squared X to the first we could just write that as X there's going to be X to the 0 which is just going to be 1 and now let's think about why this is going to start out y to the 0 which is just 1 so I'm not going to write it then it's going to be Y to the first power y squared Y to the third power Y to the fourth power Y to the fifth power Y to the sixth power and then Y to the seventh power you can verify you got it right because for each term the exponents should add up to 7 you see that even here this is X to the first times y to the sixth those add up to be seven now let's get to the interesting part which is actually calculating the coefficient and the algorithm is for each term right over here so let's just start we know the coefficient right over here is going to be one and actually let me write that down the coefficient over here is going to be one for each term the coefficient right over here let me the coefficient I'm going to try to color code it so we can see it the coefficient is going to be the exponent of the previous term so the exponent of the previous term in this case is the 7 the exponent of the previous term times the coefficient of the previous term times the coefficient of the previous term divided by divided by which term that actually was so divided by the term so that was the first term so now the coefficient on the second term is 7 times 1 divided by 7 which is going to be equal to which is going to be equal to 7 which is going to be equal to 7 now what about this one well we use the exact same process the exact same process it is going to be it is going to be the exponent on the X term I guess you could say the hot the the the exponent on X I guess we could go with the exponent on the X which is 6 times the coefficient of the previous term so times 7 so we're taking the hot this the X the X power times the coefficient of the previous term so times 7 so the X part of the previous term times the coefficient of the previous term divided by the actual I guess you could say index of the previous term so divided by 2 so what is that going to be so this is equivalent to 3 times 7 so this is going to be equal to 21 and now let's go to this term exact same idea go to the previous term what's our exponent on X it is 5 let's multiply it times the coefficient so let's multiply it times 21 and then let's divide it by which term that is so that was the third term the third term and so this is going to be let's see 5 times 21 over 3 7 so this is going to be 35 5 times 7 and we can keep going or we can recognize that there's a symmetry here if this is 1 then the last term is also going to be one if this is 7 this is 7 then the second is the second term of 7 then the second to last term is 7 if this term is 20 with the third term is 21 then the third term to the last is 21 and then if the fourth term is 35 then the fourth from the last is 35 and just like that we have figured out the expansion of X plus y X plus y to the seventh power pretty neat in my mind