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Q.
If coefficient of $x^{n}$ in $(1+x)^{101}\left(1-x+x^{2}\right)^{100}$ is nonzero, then $n$ can not be of the form
Binomial Theorem
Solution:
$(1+x)^{101}\left(1-x+x^{2}\right)^{100}=(1+x)\left(1+x^{3}\right)^{100}$
$=(1+x)\left( C _{0}+C_{1} x^{3}+C_{2} x^{6}+\ldots+C_{100} x^{300}\right)$
Clearly in this expression $x^{\lambda}$ will be present if $\lambda=3 t$,
or $\lambda=3 t+1$
So, $\lambda$ can not be of the form $3 t+2$.