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I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$.

I suppose I could use the counting rules in probability, perhaps combination= ${{n} \choose {r}}=\frac{n!}{r!(n-r!)}$.

I want to see an actual proof behind this equation. Does anyone have any ideas?

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    BTW This is sometimes called **Pascal's rule** http://www.proofwiki.org/wiki/Pascal%27s_Rule2011-12-10

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