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Thus, starting with the right-hand side of the equation:
\begin{align*}
{n-1 \choose k-1} + {n-1 \choose k} \amp = \frac{(n-1)!}{(n-k)!(k-1)!}+ \frac{(n-1)!}{(n-1-k)!\,k!}\\
\amp = \frac{(n-1)!k}{(n-k)!\,k!} + \frac{(n-1)!(n-k)}{(n-k)!\,k!}\\
\amp = \frac{(n-1)!(k+n-k)}{(n-k)!\,k!}\\
\amp = \frac{n!}{(n-k)!\, k!}\\
\amp = {n \choose k}\text{.}
\end{align*}
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