The coefficient of $x^{2}$ in the expansion of the product $(2 - x^{2}) \cdot ((1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6})$ is

106

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107

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155

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108

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Solution

The correct option is D $106$
Let $a = ((1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6})$

$∴$ Coefficient of $x^{2}$ in the expansion of the product $(2 - x)^{2} ((1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6})$

$= 2 (Coefficient of x^{2} in a) - 1 (Constant of expansion)$

In the expansion of $((1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6})$ ,

Constant $= 1 + 1 = 2$

Coefficient of $x^{2} = Coefficient of x^{2} in (^{6} C_{0} (1 + 2 x)^{6} (3 x^{2})^{0}) + Coefficient of x^{2} in (^{6} C_{1} (1 + 2 x)^{5} 3) -^{6} C_{1} (4)$

$=^{6} C_{2} 4 + 6 \times 3 - 24$

$= 6 = + 18 - 24 = 54$

Then, coefficient of $x^{2}$ in $(2 - x)^{2} ((1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6})$

$= 2 \times 54 - 1 (2) = 108 - 2 = 106$

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