Question -
Answer -
By using the formula,
P (n, r) = n!/(n – r)!
P (n, n) = n!/(n – n)!
= n!/0!
= n! [Since, 0! = 1]
Consider 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! [Since, 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[Since, P (n, n) = n!]
= P(n+1, n+1) – 1
= RHS
Hence Proved.