$\binom{n}{k} + \binom{n}{k-1} = \frac{n!}{k!(n-k)!} + \frac{n!}{(k-1)!(n-k+1)!} = \frac{n!}{k!(n-k)!} + \frac{n!}{k!(n-k)!}\frac{k}{n-k+1} = \frac{n!}{k!(n-k)!}(1+\frac{k}{n-k+1}) = \frac{n!}{k!(n-k)!}\frac{n+1}{n-k+1} = \frac{(n+1)!}{k!(n+1-k)!}=\binom{n+1}{k}$\

Note

$n!(n+1) = (n+1)!$

$(n-1)!=\frac{n!}{n}$