1*1!+2*2!+3*3!+...+10*10!
= 1*1!+2*2!+3*3!+...+10*10!+(1!+2!+3!+...+10!) -(1!+2!+3!+...+10!)
=2*1!+3*2!+4*3!+...+10*9!+11*10!-(1!+2!+3!+...+10!)
=2!+3!+4!+...+10!+11!-(1!+2!+3!+...+10!)
=11!-1