Question

Prove that P(1,1)+2×P(2,2)+3×P(3,3)........n×P(n,n)=P(n+1,n+1)-1
Question of Permutatio

Answer 1

L.H.S.=1.1P1+2.2P2+3.3P3+.......+n.nPnAs nPn=n!, henceL.H.S.=1.1!+2.2!+3.3!.......+n.n!=(1.1!+1)+2.2!+3.3!.......+n.n!−1=2!+2.2!+3.3!.......+n.n!−1=2!(1+2)+3.3!.......+n.n!−1=3,2!+3.3!.......+n.n!−1=3!+3.3!.......+n.n!−1=3!(3+1)+......+n.n!−1=4.3!+......+n.n!−1=4!+......+n.n!−1Similarly L.H.S. will reduce toL.H.S.=n!+n.n!−1=n!(n+1)−1=(n+1)!−1⇒L.H.S.==n+1Pn+1−1=R.H.S. (Hence proved)