## Two permutation problems

I'm trying to give combinatorial (not algebraic!) proofs of the following statements:

(1) $P_r^{n+1}=r!+r(P_{r-1}^n+P_{r-1}^{n-1}+...+P_{r-1}^r)$

(2) $P_r^n=\frac{n}{n-r}P_r^{n-1}$

where $P_r^n$ denotes the number of $r$-permutations of elements from an $n$-element set. In other words, if we have an $n$-element set $A=\{x_1,...,x_n\}$ then one $r$-permutation would be an ordered $r$-tuple of different elements from $A$, i.e. $(x_1,...,x_r),$such that $x_i \in A, x_i \neq x_j$ whenever $i \neq j$.

Any help would be greatly appreciated!