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  4. The coefficient of x256 in the expansion of (1-x)101(x2+x+1)100 is:

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Q. The coefficient of $x^{256}$ in the expansion of $(1-x)^{101}\left(x^{2}+x+1\right)^{100}$ is:

JEE MainJEE Main 2021Binomial Theorem

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) $
$=\left(1-x^{3}\right)^{100}-x\left(1-x^{3}\right)^{100}$
Required coefficient $(-1) \times\left(-{ }^{100} C_{85}\right)$
$={ }^{100} C_{85}={ }^{100} C_{15}$