A) \[\frac{(2n)!}{{{(n!)}^{2}}}\]
B) \[\frac{(2n)!}{(n-1)!(n+1)!}\]
C) \[\frac{(2n)!}{(n-2)!(n+2)!}\]
D) none of these
Correct Answer: C
Solution :
[c] \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x-{{C}_{2}}{{x}^{2}}+{{C}_{3}}{{x}^{3}}...+{{C}_{n-1}}{{x}^{n-1}}+{{C}_{n}}{{x}^{n}}...(1)\] \[{{(1+x)}^{n}}={{C}_{0}}{{x}^{n}}+{{C}_{1}}{{x}^{n-1}}+{{C}_{2}}{{x}^{n-2}}+...+{{C}_{n-1}}x+{{C}_{n}}...(2)\] Multiplying Eqs. (1) and (2) and equating the coefficient of \[{{x}^{\pi -2}}\], we get \[{{C}_{0}}{{C}_{2}}+{{C}_{1}}{{C}_{3}}+{{C}_{2}}{{C}_{4}}+...+{{C}_{n-2}}{{C}_{n}}\] =Coefficient of \[{{x}^{n-2}}\]in \[{{({{1}^{+}}x)}^{2n}}\] =\[^{2n}{{C}_{n-2}}\] \[=\frac{(2n)!}{(n-2)!(n+2)!}\]
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