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  4. The coefficient of x2 in the expansion of (1 - x + 2 x2)(x + (1/x))10 is

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Q. The coefficient of $x^{2}$ in the expansion of $\left(1 - x + 2 x^{2}\right)\left(x + \frac{1}{x}\right)^{10}$ is

NTA AbhyasNTA Abhyas 2020Binomial Theorem

Solution:

The required coefficient is the coefficient of $x^{12}$ in $\left(1-x+2 x^{2}\right)\left(x^{2}+1\right)^{10}$
$=\left(1-x+2 x^{2}\right)\left(1+{ }^{10} C_{1} x^{2}+{ }^{10} C_{2} x^{4}+\ldots \ldots+{ }^{10} C_{9} x^{18}+x^{20}\right)$
Coefficient of $x^{12}$ is
$1 \times{ }^{10} C_{6}+2 \times{ }^{10} C_{5}=\frac{10 \times 93 \times 8 \times 7}{4 \times 3 \times 2}+\frac{2 \times 10 \times \times 93 \times 82 \times 7 \times 6}{5 \times 4 \times 3 \times 2}$
$=210+504$
$=714$