How to combine the fraction over the common denominator?

Question

How to combine the fractions on the righthand side over the common denominator:

$$\frac{(n+1)!n!}{k!(k-1)!(n-k+1)!}=\frac{(n+k)n!(n-1)!}{k!(k-1)!(n-k)!}+\frac{n!(n-1)!}{(k-1)!(k-2)!(n-k+1)!}$$

Answer 1

Hint:

$$\begin{align} \color{red}{k!}\cdot \color{blue}{(k-1)!}\cdot \color{green}{(n-k)!} &= \color{red}{k(k-1)!}\cdot\color{blue}{(k-1)(k-2)!}\cdot\color{green}{\dfrac{(n-k+1)!}{n-k+1}}\\ &= \dfrac{\color{red}{k}\color{blue}{(k-1)}}{\color{green}{n-k+1}} \color{red}{(k-1)!}\color{blue}{(k-2)!}\color{green}{(n-k+1)!} \end{align}$$

Answer 2

Hint: $$k!=k\cdot(k-1)!\\(k-1)!=(k-1)\cdot(k-2)!\\(n-k+1)!=(n-k+1)\cdot(n-k)!$$