Question

Find n if ²ⁿC₃ : ⁿC₂ = 52 : 3

Answer 1

²ⁿC₃ : ⁿC₂ = 52 : 3

∴ `(""^(2"n")"C"_3)/(""^"n""C"_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)!)/(3 xx 2 xx 1 xx (2"n" - 3)!) xx (2 xx 1 xx ("n" - 2)!)/("n"("n" - 1)("n" - 2)!) = 52/3`

∴ `(2 xx "n" xx (2"n" - 1)2("n" - 1))/(3 xx 2 xx 1) xx (2 xx 1)/("n"("n" - 1)) = 52/3`

∴ `(4(2"n" - 1))/3 = 52/3`

∴ 4(2n – 1) = 52

∴ 2n – 1 = `52/4`

∴ 2n – 1 = 13

∴ 2n = 14

∴ n = 7.