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By the definition of \({n \choose k}\text{,}\) we have
\begin{equation*}
{n-1 \choose k-1} = \frac{(n-1)!}{(n-1-(k-1))!(k-1)!} = \frac{(n-1)!}{(n-k)!(k-1)!}
\end{equation*}
and
\begin{equation*}
{n-1 \choose k} = \frac{(n-1)!}{(n-1-k)!k!}\text{.}
\end{equation*}
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