Question

Show that (n + 1) (ⁿP_r) = (n – r + 1) [⁽ⁿ⁺¹⁾P_r]

Answer 1

L.H.S. = (n + 1) (ⁿP_r) 

= `("n"+ 1)("n"!)/(("n" - "r")!)`

= `(("n" + 1)!)/(("n" - "r")!)`    ...(1)

R.H.S. = (n – r + 1) [⁽ⁿ⁺¹⁾P_r]

= `("n" - "r" + 1)* (("n" + 1)!)/(("n" + 1 - "r")!)`

= `("n" - "r" + 1)* (("n" + 1)!)/(("n" - "r" + 1)!)`

= `("n" - "r" + 1)* (("n" + 1)!)/(("n" - "r" + 1)("n" - "r")!)`

= `(("n" + 1)!)/(("n" - "r")!)`    ...(2)

From (1) and (2), we get, L.H.S. = R.H.S.

Hence, (n + 1) (ⁿP_r) = (n – r + 1) [⁽ⁿ⁺¹⁾P_r]