To prove: P(1, 1) + 2. P(2, 2) + 3 . P(3, 3) + … + n . P(n, n) = P(n + 1, n + 1) – 1

We know,

Take L.H.S.:

1. P(1, 1) + 2. P(2, 2) + 3. P(3, 3) + … + n . P(n, n)

= 1.1! + 2.2! + 3.3! +………+ n.n!

{∵ P(n, n) = n!}

= (2! – 1!) + (3! – 2!) + (4! – 3!) + ……… + (n! – (n – 1)!) + ((n+1)! – n!)

= 2! – 1! + 3! – 2! + 4! – 3! + ……… + n! – (n – 1)! + (n+1)! – n!

= (n + 1)! – 1!

= (n + 1)! – 1

{∵ P(n, n) = n!}

= P(n+1, n+1) – 1

= R.H.S

Hence Proved