Question

The coefficient of xⁿ in the expansion of (1 + x)²ⁿ and (1 + x)^2n–1 are in the ratio ______.

Options

• 1 : 2

• 1 : 3

• 3 : 1

• 2 : 1

Answer 1

The coefficient of xⁿ in the expansion of (1 + x)²ⁿ and (1 + x)^2n–1 are in the ratio 2 : 1

Explanation:

General Term `"T"_(r + 1) = ""^"n""C"_r  x^(n - r) y^r`

In the expansion of (1 + x)²ⁿ

We get `"T"_(r + 1) = ""^(2n)"C"_r  x^r`

To get the coefficient of xⁿ

Put r = n

∴ Coefficient of xⁿ = ²ⁿC_n

In the expansion of (1 + x)^2n–1 

We get `"T"_(r + 1) = ""^(2n - 1)"C"_r x^r`

∴ Coefficient of xⁿ = `""^(2n - 1)"C"_(n - 1)`

The required ratio is `(""^(2n)"C"_n)/(""^(2n - 1)"C"_(n - 1))`

= `((2n!)/(n!(n!)))/(((2n - 1)!)/((n - 1)!(2n - 1 - n + 1)!))`

= `((22n!)/(n!n!))/(((2n - 1)!)/((n - 1)(n!))`

= `(2n!)/(n!n!) xx ((n - 1)!*n!)/((2n - 1)!)`

= `(2n(2n - 1)!)/(n!n(n - 1)!) xx ((n - 1)!*n!)/((2n - 1)!)`

= `2/1`

= 2 : 1