Question

Find n, if: `((2"n")!)/(7!(2"n" - 7)!) : ("n"!)/(4!("n" - 4)!)` = 24:1

Answer 1

`((2"n")!)/(7!(2"n" - 7)!) : ("n"!)/(4!("n" - 4)!)` = 24:1

∴ `((2"n")!)/(7!(2"n" - 7)!) : (4!("n" - 4)!)/("n"!)` = 24

`∴((2"n")(2"n" - 1)(2"n" - 2)(2"n" - 3)(2"n" - 4)(2"n" - 5)(2"n" - 6)(2"n" - 7)!)/(7xx6xx5xx4!(2"n"-7)!)xx(4!("n" - 4)!)/("n"("n" -1)("n" - 2)("n" - 3)("n" - 4)!)=24`

`∴((2"n")(2"n" - 1)(2"n" - 2)(2"n" - 3)(2"n" - 4)(2"n" - 5)(2"n" - 6))/(7 xx 6 xx 5)xx1/("n"("n" -1)("n" - 2)("n" - 3))=24`

`∴((2"n")(2"n" - 1)2("n" - 2)(2"n" - 3)2("n" - 2)(2"n" - 5)2("n" - 3))/(7 xx 6 xx 5)xx1/("n"("n" -1)("n" - 2)("n" - 3))=24`

∴ `(16(2"n" - 1)(2"n" - 3)(2"n" - 5))/(7xx6xx5)` = 24

∴ (2n – 1) (2n – 3) (2n – 5) = `(24xx7xx6xx5)/16`
∴ (2n – 1) (2n – 3) (2n – 5) = 9 × 7 × 5
Comparing on both sides, we get
∴ 2n – 1 = 9
∴ n = 5