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Question
Find n and r if `""^"n""P"_"r"` = 720 and `""^"n""C"_("n" - "r")` = 120
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Solution
`""^"n""P"_"r"` = 720
∴ `("n"!)/(("n" - "r")!)` = 720 ...(i)
Also, `""^"n""C"_("n" - "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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