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