Is there a short expression for this?

Question

Is there a closed form expression for $$\Bigg(\binom{n}{k-1}+\binom{n-1}{k-1}+\dots+\binom{k-1}{k-1}\Bigg)(k-1)!= \sum_{i=0}^{n-k+1}\frac{(n-i)!}{(n-k+1-i)!},$$ $$\Bigg(\binom{n}{k-1}+\binom{n-1}{k-1}+\dots+\binom{k}{k-1}\Bigg)(k-1)!= \sum_{i=0}^{n-k}\frac{(n-i)!}{(n-k+1-i)!}?$$

Answer 1

Using the Hockey-Stick Identity, we have $\displaystyle\sum_{r = k-1}^{n}\dbinom{r}{k-1} = \dbinom{n+1}{k}$.

Hence, $(k-1)! \cdot \displaystyle\sum_{r = k-1}^{n}\dbinom{r}{k-1} = (k-1)! \cdot \dbinom{n+1}{k} = \dfrac{(k-1)!(n+1)!}{k!(n+1-k)!} = \dfrac{(n+1)!}{k(n+1-k)!}$.

Simplify as needed.