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

107

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108

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155

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106

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Solution

The correct option is D $106$
$(2 - x^{2}) [(1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6}] = 2 [(1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6}] - x^{2} [(1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6}] \Rightarrow coefficient of x^{2} = 2 \cdot coefficient of x^{2} in [(1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6}] - constant term in [(1 + 2 x + 3 x^{2})^{6} + (1 - 4 x^{2})^{6}] (1 + 2 x + 3 x^{2})^{6} = 6 \sum r = 0^{6} C_{r} (2 x + 3 x^{2})^{r} =^{6} C_{0} +^{6} C_{1} (2 x + 3 x^{2}) +^{6} C_{2} (2 x + 3 x^{2})^{2} + . . . . . \Rightarrow coefficient of x^{2} = 2 (18 + 60 - 24) - 2 = 106$

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