Solution
= Solution
For $0\leq k\leq n$, the <binomial coefficient> is
$$
\binom nk=\frac{n!}{k!(n-k)!}.
$$
When $1\leq k\leq n-1$,
$$
\begin{aligned}
\binom{n-1}k+\binom{n-1}{k-1}
&=\frac{(n-1)!}{k!(n-1-k)!}
+\frac{(n-1)!}{(k-1)!(n-k)!}\\
&=\frac{(n-1)![(n-k)+k]}{k!(n-k)!}\\
&=\boxed{\binom nk}.
\end{aligned}
$$
This is <Pascal's identity>.