A) \[^{30}{{C}_{10}}\]
B) \[^{30}{{C}_{25}}\]
C) \[1\]
D) None of these
Correct Answer: B
Solution :
We have, the expression \[{{(1+{{x}^{2}})}^{40}}{{\left( {{x}^{2}}+2+\frac{1}{{{x}^{2}}} \right)}^{-5}}\] \[={{(1+{{x}^{2}})}^{40}}{{\left\{ {{\left( x+\frac{1}{x} \right)}^{2}} \right\}}^{-5}}\] \[={{(1+{{x}^{2}})}^{40}}{{\left( \frac{{{x}^{2}}+1}{x} \right)}^{-10}}\] \[={{x}^{10}}{{(1+{{x}^{2}})}^{30}}\] Now, coefficient of \[{{x}^{20}}\] in the expansion of\[{{x}^{10}}{{(1+{{x}^{2}})}^{20}}\] = coefficient of \[{{x}^{10}}\] in the expansion of\[{{(1+{{x}^{2}})}^{30}}\] = coefficient of \[{{x}^{10}}\] in \[(1{{+}^{30}}{{C}_{1}}x{{+}^{30}}{{C}_{2}}{{x}^{4}}{{+}^{30}}{{C}_{x}}^{6}{{+}^{30}}{{C}_{4}}{{x}^{8}}\] \[{{+}^{30}}{{C}_{5}}{{x}^{10}}+...)\] \[{{=}^{30}}{{C}_{5}}\] \[{{=}^{30}}{{C}_{25}}\] \[(\because \,{{\,}^{n}}{{C}_{r}}{{=}^{n}}{{C}_{n-r}})\]
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