If coefficients of $x^{n}$ in $(1 + x)^{101} (1 - x + x^{2})^{100}$ is non-zero then $n$ cannot be of the form

3 t + 1

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3 t

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3 t + 2

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4 t + 1

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Solution

The correct option is D $3 t + 2$
Given expression is $(1 + x)^{101} (1 - x + x^{2})^{100}$ , Let it be $y$
$y = (1 + x) ((1 + x) (1 - x + x^{2}))^{100} = (1 + x) (1 + x^{3})^{100}$
We know that $(1 + x^{3})^{100} = 1 + a x^{3} + b x^{6} + c x^{9} + . . . .$
$\Rightarrow y = (1 + x) (1 + x^{3})^{100} = 1 + x + a x^{3} + a x^{4} + b x^{6} + b x^{7} + . . . .$
We can observe that $x^{3 t + 2}$ terms are missing , which means the coefficients of $x^{3 t + 2}$ terms are $0$
Therefore for the coefficient of $x^{n}$ not to be zero , $n$ cannot be $3 t + 2$

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