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Question
Find n, if: `("n"!)/(3!("n" - 3)!) : ("n"!)/(5!("n" - 7)!)` = 1 : 6
Solution
`("n"!)/(3!("n" - 3)!) : ("n"!)/(5!("n" - 7)!)` = 1 : 6
∴ `("n"!)/(3!("n" - 3)!) xx (5!("n" - 7)!)/("n"!) = 1/6`
∴ `(5!("n" - 7)!)/(3!("n" - 3)!) = 1/6`
∴ `(5 xx 4 xx 3! xx ("n" - 7)!)/(3! xx ("n" - 3)("n" - 4)("n" - 5)("n" - 6)("n" - 7)!) = 1/6`
∴ `(5 xx 4)/(("n" - 3)("n" - 4)("n" - 5)("n" - 6)) =1/6`
∴ (n – 3)(n – 4)(n – 5)(n – 6) = 6 × 5 × 4
∴ (n – 3)(n – 4)(n – 5)(n – 6) = 5 × 4 × 3 × 2
∴ (n – 3)(n – 4)(n – 5)(n – 6) = (8 – 3)(8 – 4)(8 – 5)(8 – 6)
∴ n = 8
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