Prove that:
1 . P (1, 1) + 2 . P (2, 2) + 3 . P (3, 3) + ... + n . P (n, n) = P (n + 1, n + 1) − 1.

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Solution

1.P (1, 1) + 2. P (2, 2) + 3. P (3, 3) + ... + n . P (n, n) = P (n + 1, n + 1) − 1
P (n,n) = n!
1.1! + 2.2! + 3.3! ......+ n.n! = (n+1)! − 1
LHS = 1.1! + 2.2! + 3.3! ......+ n.n!
$= \sum_{r = 1}^{n} r . r! = \sum_{r = 1}^{n} [(r + 1) - 1] r! = \sum_{r = 1}^{n} [(r + 1) r! - r!] = \sum_{r = 1}^{n} {(r + 1)! - r!} = (2! - 1!) + (3! - 2!) + . . . [(n + 1)! - n!] = [(n + 1)! - 1!] = [(n + 1)! - 1] = RHS Hence, proved .$

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