Question

If ⁿP_r = 720 and ⁿC_r = 120, find n, r

Answer 1

Given ⁿP_r = 720, If ⁿC_r = 120

`("n"!)/(("n" - "r")!)` = 720

`("n"!)/("r"!("n" - "r")!)`  120

`(("n"!)/(("n" - "r")!))/(("n"!)/("r"("n" - "r")!)) = 720/120`

`("n"!)/(("n" - "r")!) xx ("r"!("n" - "r")!)/("n"!)` = 6

r! = 6

r! = 3 × 2 × 1 = 3!

r = 3

Substituting r = 3 in `("n"!)/(("n" - "r")!)` = 720

`("n"!)/(("n" - 3)!)` = 720

`("n"("n" - 1)("n" - 2)("n" - 3)!)/(("n" - 3)!)` = 720

n (n – 1)(n – 2) = 720

n(n – 1)(n – 2) = 10 × 9 × 8

n = 10

∴ r = 3, n = 10