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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

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Question

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

Options

$\frac{n!}{[(n - 1)! (n + 1)!]}$
$\frac{(2 n)!}{[(n - 1)! (n + 1)!]}$
$\frac{(2 n)!}{(2 n - 1)! (2 n + 1)!}$
none of these

MCQ

Solution

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

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

$=^{n} C_{0} \times^{n} C_{1} +^{n} C_{1} \times^{n} C_{2}$

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

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

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

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Introduction of Binomial Theorem

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Chapter 18: Binomial Theorem - Exercise 18.4 [Page 47]

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RD Sharma Mathematics [English] Class 11

Chapter 18 Binomial Theorem
Exercise 18.4 | Q 20 | Page 47

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