Here I will show how to derive the binomial coefficients

using generating functions (using the approach from Wilf’s *generatingfunctionology*). For each nonnegative integer and integer with , suppose is the number of -element subsets of . Since each such subset of either contains or does not, we have the recurrence

for , with for . For each , define the generating function

Multiplying each side of the recurrence by and summing over we get

and hence

so that

for . Since , it follows that

Since is a polynomial, it is equal to its Taylor expansion:

It is clear that

where

Therefore , so that the coefficient of in the Taylor expansion is

This is the familiar expression for , and so

the same result as that from the binomial theorem!

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Cool way to look at it!

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Wow, people read this blog? 😛

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