# The Coefficient of 1 X in the Expansion of ( 1 + X ) N ( 1 + 1 X ) N Is(A) N ! [ ( N − 1 ) ! ( N + 1 ) ! ](B) ( 2 N ) ! [ ( N − 1 ) ! ( N + 1 ) ! ] (C) ( 2 N ) ! ( 2 N − 1 ) ! ( 2 N + 1 ) ! - Mathematics

MCQ

The coefficient of  $\frac{1}{x}$  in the expansion of $\left( 1 + x \right)^n \left( 1 + \frac{1}{x} \right)^n$ is

#### Options

•  $\frac{n !}{\left[ \left( n - 1 \right) ! \left( n + 1 \right) ! \right]}$

• $\frac{\left( 2n \right) !}{\left[ \left( n - 1 \right) ! \left( n + 1 \right) ! \right]}$

•  $\frac{\left( 2n \right) !}{\left( 2n - 1 \right) ! \left( 2n + 1 \right) !}$

•  none of these

#### Solution

$\frac{\left( 2n \right) !}{\left[ \left( n - 1 \right) ! \left( n + 1 \right) ! \right]}$

$\text{ Coefficient of } \frac{1}{x}\text{ in the given expansion = Coefficient of 1 in } (1 + x )^n \times \text{ Coefficient of} \frac{1}{x}in \left( 1 + \frac{1}{x} \right)^n + \text{ Coefficient of x in } (1 + x )^n \times \text{ Coefficient of } \frac{1}{x^2} \text{ in } \left( 1 + \frac{1}{x} \right)^n$

$=^{n}{}{C}_0 \times ^{n}{}{C}_1 +^{n}{}{C}_1 \times ^{n}{}{C}_2$

$= n + n \times \frac{n!}{2\left( n - 2 \right)!}$

$= n + n\frac{n\left( n - 1 \right)}{2}$

$=$ $\frac{\left( 2n \right) !}{\left[ \left( n - 1 \right) ! \left( n + 1 \right) ! \right]}$

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Q 20 | Page 47