[Music] do [Music] our story begins in an ancient forest in the remote town of east vandermond colin and sylvia two college students on spring break are hiking through the forest why did you want to come all the way here sylvia we're in the middle of nowhere we should be studying for our probability course perhaps we are what are you doing these humans won't understand our sacred identities patience my friend they will learn what is this colin and sylvia don't know what to make of the scroll they have found that's impossible it looks like they just added the a and b to get a plus b and added the k and c minus k to get c what kind of bird brain would come up with a formula like this it looks mysterious and beautiful but that hardly means it isn't true let's try some examples and see if we can prove or disprove it colin remembers from stat 110 how a binomial coefficient n choose k can be defined in terms of factorials n factorial over n minus k factorial k factorial he tries a couple examples on his calculator it is true in those examples so he proceeds to work on a proof using algebra it's more of a mystery than ever i wonder if we'll ever know if this formula is always true or how anyone could have come up with it actually i can prove it in a few sentences with no algebra needed huh look at the birds they will show you sylvia reminds colin that they learned in stat 110 that the binomial coefficient and choose k can also be defined by a story it's the number of ways to choose a committee of size k from n people where the order in which people are chosen doesn't matter it's the number of subsets of size k for a set of size n [Music] it's the number of flocks of k birds that can be formed by choosing birds from a flock of n birds i see peacocks and two cans flying around trying to get on the top branches but what does that have to do with the formula on the scroll let's say there are a peacocks and b two cans vying for the six top branches the number of possibilities for which birds get to perch on top branches is a plus b choose six that's the right hand side when c is six okay but what about the left hand side the left-hand side just breaks this into cases if there is one peacock perched on a top branch there must be five two cans perched on top branches if there are two peacocks there must be four two cans if there are k peacocks there must be c minus k two cans so count how many possibilities there are for which k peacocks and which c minus k two cans are on top branches then sum up the cases since we want the total number of possibilities not just the number for one value of k the formula on the scroll is called vandermonde's identity sylvia has just discovered a story proof of this identity she has given it an interpretation rather than doing tedious or intractable algebra both sides of the identity count the same thing so they must be equal sylvia gave her proof for the case c equal to six because she was looking at a tree with six top branches but the same idea works for any value of c van der maan's identity is a handy result that often comes up in problems involving binomial coefficients thinking about it in terms of a story makes it easier to remember and understand the moral of the story is to look for a story [Music] you