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Question
Find n and r if nPr = 720 and nCn–r = 120
Solution
nPr = 720
∴ `("n"!)/(("n" - "r")!)` = 720 ...(i)
Also, nCn–r = 120
∴ `("n"!)/(("n" - "r")!("n" - "n" + "r")!)` = 120
∴ `("n"!)/("r"!("n" - "r")!)` = 120 ...(ii)
Dividing (i) by (ii), we get
`(("n"!)/(("n" - "r")!))/(("n"!)/("r"!("n" - "r")!)) = 720/120`
∴ r! = 6
∴ r = 3
Substituting r = 3 in (i), we get
`("n"!)/(("n" - 3)!)` = 720
∴ `("n"("n" - 1)("n" - 2)("n" - 3)!)/(("n" - 3)!)` = 720
∴ n(n – 1) (n – 2) = 10 × 9 × 8
∴ n = 10
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