[ \dfrac{(n+2)!}{(r+2)!} + \dfrac{(n+1)!}{(r+1)!} = \dfrac{(n+2)!}{(r+2)!} + \dfrac{(n+1)!(r+2)}{(r+2)!} = \dfrac{(n+2)! + (n+1)!(r+2)}{(r+2)!} = \dfrac{(n+1)!\bigl[(n+2) + (r+2)\bigr]}{(r+2)!} = \dfrac{(n+1)!(n+r+4)}{(r+2)!}. ]

Answer: ( \dfrac{(n+1)!(n+r+4)}{(r+2)!} )