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

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

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

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

Any help would be greatly appreciated!