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I'm trying to answer the following question:

By counting a set in two ways, prove the following formula

$$ \sum_{i=0}^k {m \choose i}{n \choose k-i} = {m+n \choose k} $$

I found some similar examples on this site and came to the conclusion of saying the following for the answer

We can say that $m$ represents the number of boys and $n$ represents the number of girls and that we are picking $k$ number of people from the total number of boys and girls for the right side of the equation.

For the left side, we say again that there are $m$ boys and $n$ girls and that we are picking $k$ number of boys and girls but we are picking $i$ boys and then $k-i$ girls.

This is a much as I could do with the question with my current understanding. I don't know if this constitutes as a full proof or if I have to add more to it, and if I do have to add more I don't know what I'm missing. If someone could point me in the right direction it would be greatly appreciated.

AFC
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  • Sounds like a correct proof. Each set from the right side has some number of girls in it from $0$ to $k$ and it falls into exactly one of the sets on the left side. You can write it in the language of sets, maybe that will be more clear. – Artur Riazanov Jan 20 '18 at 19:48

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