A) \[\frac{n!}{(n-1)!(n+1)!}\]
B) \[\frac{(2n)\,!}{(n-1)!(n+1)!}\]
C) \[\frac{n!}{(n-1)!(n+1)!}\]
D) None of these
Correct Answer: B
Solution :
\[{{(1+x)}^{n}}={{\,}^{n}}{{C}_{0}}+{{\,}^{n}}{{C}_{1}}x+{{\,}^{n}}{{C}_{2}}{{x}^{2}}+....+{{\,}^{n}}{{C}_{n}}{{x}^{n}}\] \[{{\left( 1+\frac{1}{x} \right)}^{n}}={{\,}^{n}}{{C}_{0}}+{{\,}^{n}}{{C}_{1}}\frac{1}{x}+{{\,}^{n}}{{C}_{2}}\frac{1}{{{x}^{2}}}+....+{{\,}^{n}}{{C}_{n}}{{\left( \frac{1}{x} \right)}^{n}}\] Obviously, required coefficient of \[\frac{1}{x}\] can be given by \[^{n}{{C}_{0}}{{\,}^{n}}{{C}_{1}}+{{\,}^{n}}{{C}_{1}}{{\,}^{n}}{{C}_{2}}+....+{{\,}^{n}}{{C}_{n-1}}^{n}{{C}_{n}}\]\[=\frac{(2n)!}{(n-1)!(n+1)!}\]
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