$$ \frac{n!}{k!(n-k)!} + \frac{n!}{(k+1)!\left(n-(k+1)\right)!} $$
$$ = \frac{ \mathbf{(k+1)} n! }{ \underbrace{\mathbf{(k+1)} k!}_{=(k+1)!} (n-k)! } + \frac{ \mathbf{(n-k)}n! }{ (k+1)! \underbrace{\mathbf{(n-k)} (n-k-1)!}_{=(n-k)!} }$$
$$ = \frac{ \mathbf{(k+1)} n! }{ \mathbf{(k+1)!}(n-k)! } + \frac{ \mathbf{(n-k)}n! }{ (k+1)! \mathbf{(n-k)!} }$$
$$ = \frac{ (k+1)n! + (n-k)n! }{ (k+1)!\underbrace{(n-k)!}_{=((n+1)-(k+1))!} }$$
$$ = \frac{ n!(k+1+n-k) }{ (k+1)!\left((n+1)-(k+1)\right)! } $$
Jetzt bist du wieder dran.