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The ratio of the coefficient of $x^{10}$ in $(1-x^2)^{10}$ and the term independent of $x$ in $(x-\large\frac{2}{x}$$)^{10} is ? \begin{array}{1 1} 1:31 \\ 1:30 \\ 1:32 \\ 1:33 \end{array} Can you answer this question? 1 Answer 0 votes General term in (-1)^r.(1-x^2)^{10} is ^{10}C_rx^{2r} For coefficient of x^{10}, 2r=10 \Rightarrow\:\:r=5 \therefore Coefficient of x^{10} in (1-x^2)^{10} is -^{10}C_5 Similarly the general term in (x-\large\frac{2}{x}$$)^{10}$ is $(-1)^r.^{10}C_r.x^{10-r}.2^r.x^{-r}$
$=(-1)^r.^{10}C_r.2^r.x^{10-2r}$
For independent term in this expansion, $10-2r=0$ or $r=5$
$\therefore\:$ The independent term in $(x-\large\frac{2}{x}$$)^{10} is -^{10}C_5.2^5 \Rightarrow\: The ratio of coeff. of x^{10} in (1-x^2)^{10} and the independent term in (x-\large\frac{2}{x}$$)^{10}$ is
$-^{10}C_5\: :\:-^{10}C_5.2^5=1:32$