Coefficient of frac{1}{x} in expansion of {left( {1 + x} right)^n}{left( {1 + frac{1}{x}

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7.Binomial Theorem

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

A

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

B

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

C

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

D

None of these

Solution

Simplifying, we get $\frac{(1+x)^{2 n}}{x^{n}} \ldots( i )$

$T_{r+1}={ }^{2 n} C_{r} x^{r-n}$

Hence coefficient of $x^{-1}$

$=2 n C_{n-1}$

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

Standard 11

Mathematics

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