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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

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

     
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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?
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APPEARS IN

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