\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} \end{align*}
in-context