If coefficient 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 C $3 t + 2$
${(1 + x)}^{101} {(1 - x + x^{2})}^{100} = (1 + x) {(1 + x^{3})}^{100} = (1 + x) (^{n} C_{0} +^{n} C_{1} x^{3} +^{n} C_{2} x^{6} + . . . +^{n} C_{100} x^{300})$
Clearly, in the expression $x^{λ}$ will be present if $λ = 3$ , or $λ = 3 t + 1$
So, $λ$ cannot be of the form $3 t + 2$

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