# If I have (n choose k) how can I rewrite it to be something that ends in “choose k+1”?

See title for question; just trying to rewrite a combinatoric

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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 $$\binom{n}{k}=\frac{k+1}{n-k}\binom{n}{k+1}.$$