To show it is true $r\times r!$ can be algebraically decomposed as $(r+1)!-r!$
But I am trying to think of a combinatorial proof. If $S$ is the set of all permutations of $\{1,2,\cdots ,n\}$ then I need a definition for the mutually non overlapping subsets $S_r$ whose union is $S$. So I will have $|S|=\sum|S_r|$. I am unable to think of such a decomposition.