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

Question

The coefficient of xn in the expansion of (1 + x)2n and (1 + x)2n-1 are in the ratio (A) 1: 2 (B) 1: 3 (C) 3: 1 (D) 2: 1

Tardigrade Admin · Accepted Answer

Coefficient of xn in (1+x)2n = 2nCn and coefficient of xn in (1 + x)2n-1= 2n-1Cn ∴ Required ratio = (2nCn/2n-1Cn) = (((2n)!/n!n!)/((2n-1)!)n!(n-1)!) = 2: 1

Q. The coefficient of $x^{n}$ in the expansion of $(1 + x)^{2 n}$ and $(1 + x)^{2 n - 1}$ are in the ratio

$1 : 2$

$1 : 3$

$3 : 1$

$2 : 1$

Coefficient of $x^{n}$ in $(1 + x)^{2 n} =^{2 n} C_{n}$
and coefficient of $x^{n}$ in $(1 + x)^{2 n - 1} =^{2 n - 1} C_{n}$
$∴$ Required ratio $= \frac{^{2 n} C _{n}}{^{2 n - 1} C _{n}}$
$= \frac{\frac{( 2 n ) !}{n ! n !}}{\frac{( 2 n - 1 ) !}{n ! ( n - 1 ) !}} = 2 : 1$

Q. The coefficient of xn in the expansion of (1+x)2n and (1+x)2n−1 are in the ratio

1:2

1:3

3:1

2:1