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

We have,
LHS = 1. P(1,1)+2. P(2,2)+3. P(3,3)+...+n. P(n,n)
= $1.1 + 2.2! + 3.3! . . . . . n . n! [∵ P (n, n) = n!] = n Σ r - 1 r . r! = n Σ r - 1 [(r + 1) r! - r!] = n Σ r - 1 [(r + 1)! - r!] [∵ (r + 1) r! = (r + 1)!] = [(2! - 1!) + (3! - 2!) + (4! - 3!) . . . . + (n + 1)! - n!] = (n + 1)! = 1! = {n + 1}_{P_{n + 1}} - 1! [∵ n_{P_{n}} = n!] = P (n + 1, n + 1) - 1$
$\Rightarrow$ LHS=RHS
Hence proved.

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