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  4. The coefficient of x20 in the expansion of (1+x2)40 ⋅(x2+2+(1/x2))-5 is

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Q. The coefficient of $x^{20}$ in the expansion of $\left(1+x^{2}\right)^{40} \cdot\left(x^{2}+2+\frac{1}{x^{2}}\right)^{-5}$ is

Bihar CECEBihar CECE 2013

Solution:

$ \because\left(1+x^{2}\right) ^{40} \cdot\left(x^{2}+2+\frac{1}{x^{2}}\right)^{-5} $
$ =\left(1+x^{2}\right)^{20} \cdot x^{10} $
$=\left(1+x^{2}\right)^{30} \cdot x^{20} $
$\therefore $ Coefficient of $x^{10}$ in $\left(1+x^{2}\right)^{30} \cdot x^{10}$
$\Rightarrow $ Coefficient of $x^{10}$ in $\left(1+x^{2}\right)^{30}$ is ${ }^{30} C_{5}$ or ${ }^{30} C _{25}$