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Question
Find n if `""^(2"n")"C"_3: ""^"n""C"_2` = 52:3
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Solution
∴ `(""^(2"n")"C"_3)/(""^"n""C"_2) = 52/3`
∴ `((2"n")!)/(3!(2"n" - 3)!)÷("n"!)/(2!("n" - 2)!) = 52/3`
∴ `((2"n")!)/(3!(2"n" - 3)!) xx (2!("n" - 2)!)/("n"!) = 52/3`
`∴ ((2"n")(2"n" - 1)(2"n" - 2)(2"n" - 3)!)/(3xx2!(2"n" - 3)!) xx (2!("n" - 2)!)/("n"("n" - 1)("n" - 2)!) = 52/3`
∴ `(2"n"(2"n" - 1) xx 2("n" - 1))/3 xx 1/("n"("n" - 1)) = 52/3`
∴ `(4(2"n" - 1))/3 = 52/3`
∴ 2n – 1 = `52/3 xx 3/4`
∴ 2n – 1 = 13
∴ 2n = 14
∴ n = 7
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