Salut,
[tex]S=\sum\limits_{k=1}^n(n-k)\cdot A_n^{k-1}=\sum\limits_{k=1}^n(n-k)\dfrac{n!}{(n-k+1)!}=n!\cdot\sum\limits_{k=1}^n\dfrac{n-k+1-1}{(n-k+1)!}=\\\\=n!\cdot\sum\limits_{k=1}^n\left[\dfrac{n-k+1}{(n-k+1)!}-\dfrac{1}{(n-k+1)!}\right]=n!\cdot\sum\limits_{k=1}^n\left[\dfrac{n-k+1}{(n-k+1)(n-k)!}-\dfrac{1}{(n-k+1)!}\right]=\\\\=n!\cdot\sum\limits_{k=1}^n\left[\dfrac{1}{(n-k)!}-\dfrac{1}{(n-k+1)!}\right].\ Not\breve{a}m\ cu\ S_1=\sum\limits_{k=1}^n\left[\dfrac{1}{(n-k)!}-\dfrac{1}{(n-k+1)!}\right].\\\\S_1=\dfrac{1}{(n-1)!}-\dfrac{1}{n!}+\\\\+\dfrac{1}{(n-2)!}-\dfrac{1}{(n-1)!}+\\\\+\dfrac{1}{(n-3)!}-\dfrac{1}{(n-2)!}+\\\\+\ldots+\\\\+\dfrac{1}{1!}-\dfrac{1}{2!}+\\\\+\dfrac{1}{0!}-\dfrac{1}{1!}=1-\dfrac{1}{n!}.\\\\Suma\ din\ enun\c{t}\ S=n!\cdot S_1=n!-1.[/tex]
Green eyes.
