Everything Pascal's Triangle: The Binomial Theorem (and Generating Functions)

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The Binomial Theorem is one of the best parts of Pascal Triangle is the Binomial Theorem. It's not interesting because of it's proof, but because of the seemingly coincidental way it seamlessly connects to combinatorics. However, I don't really like coincidences, so it's time to talk about generating functions, and how the binomial theorem is merely a representative of another, larger math idea to combine algebra and combinatorics.

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