If Pm  stands for  mPm , then 1+1.P1+2.P2+3.P3+.........+n.Pn=

We have Pm= mPm=m! ∴1+1.P1+2.P2+3.P3+........+n.Pn =1+1+2.2!+3.3!+.....+n.n! =1+∑r=1nr.r!=1+∑r=1n(r+1−1)r! =1+∑r=1n{(r+1)!−r!}=1+{2!−1!+3!−2!+4!−3!+......+(n+1)!−n!}=(n+1)!