Question:
The coefficient of $x^{256}$ in the expansion of $(1-x)^{101}\left(x^{2}+x+1\right)^{100}$ is:
Correct Option: , 2
Solution:
$(1-x)^{100} \cdot\left(x^{2}+x+1\right)^{100} \cdot(1-x)$
$=\left((1-x)\left(x^{2}+x+1\right)\right)^{100}(1-x)$
$=\left(1^{3}-x^{3}\right)^{100}(1-x)$
$=\left(1-x^{3}\right)^{100}(1-x)$
$=\underbrace{\left(1-x^{3}\right)^{100}}_{\text {Notermof } x^{256}}-\underbrace{x}_{\text {We findcofficientof } x^{255}}$
Required coefficient $(-1) \times\left(-{ }^{100} \mathrm{C}_{85}\right)$
$={ }^{100} \mathrm{C}_{85}={ }^{100} \mathrm{C}_{15}$
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