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See title for question; just trying to rewrite a combinatoric

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up vote 3 down vote accepted

Suppose that $k\ne n$. Then $$\binom{n}{k}=\frac{n!}{k!(n-k)!}=(k+1)\frac{n!}{(k+1)!(n-k)!}=\frac{k+1}{n-k}\frac{n!}{(k+1)!(n-k-1)!}.$$ It follows that $$\binom{n}{k}=\frac{k+1}{n-k}\binom{n}{k+1}.$$

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