A) \[\frac{(2n)!}{(n+1)!(n+2)!}\]
B) \[\frac{(2n)!}{(n-2)!(n+2)!}\]
C) \[\frac{(2n)!}{(n)!(n+2)!}\]
D) \[\frac{(2n)!}{(n-1)!(n+2)!}\]
Correct Answer: B
Solution :
We have, \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}....+{{C}_{n}}{{x}^{n}}\] \[{{\left( 1+\frac{1}{x} \right)}^{n}}={{C}_{0}}+{{C}_{1}}.\frac{1}{x}+{{C}_{2}}.\frac{1}{{{x}^{2}}}+...+{{C}_{n}}\left( \frac{1}{{{x}^{n}}} \right)\] on multiplying both expansions, we get \[\frac{{{(1+x)}^{2n}}}{{{x}^{n}}}=\sum{C_{0}^{2}+x\sum{{{C}_{0}}{{C}_{1}}+{{x}^{2}}\sum{{{C}_{0}}{{C}_{2}}+....}}}\]\[+{{x}^{r}}\sum{{{C}_{0}}{{C}_{r}}+.....}\] The various sigma are the coefficient of \[{{x}^{0}},x,{{x}^{2}},.....,{{x}^{r}}\] in L.H.S. \[\frac{{{(1+x)}^{2n}}}{{{x}^{n}}}\]or coefficient of \[{{x}^{n}},{{x}^{n+1}},{{x}^{n+2}},.....,{{x}^{n+r}}\] in the expansion of \[{{(1+x)}^{2n}}\] which occur in \[{{T}_{n+1,}}{{T}_{n+2}},....\] and are \[^{2n}{{C}_{n}}{{,}^{2n}}{{C}_{n+1}}{{,}^{2n}}{{C}_{n+2}}{{....}^{2n}}{{C}_{n+r}}\]etc. \[^{\,2n}{{C}_{n+2}}=\frac{(2\,n\,)\,!}{(n-2)\,!\,(n+2)\,!}\]
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