The coefficient of $x^{256}$ in the expansion of $(1 - x)^{101} (x^{2} + x + 1)^{100}$ is

^{100} C_{15}

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^{100} C_{16}

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-^{100} C_{16}

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-^{100} C_{15}

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Solution

The correct option is A $^{100} C_{15}$
$(1 - x)^{101} (x^{2} + x + 1)^{100}$
$\Rightarrow (1 - x) ((1 - x) (1 + x + x^{2}))^{100}$
$\Rightarrow (1 - x) (1 - x^{3})^{100}$
General term in $(1 - x^{3})^{100}$ is $^{100} C_{r} {(- x^{3})}^{r} =^{100} C_{r} {(- x)}^{3 r}$

$∴ x^{256}$ occurs if $3 r = 256 or 3 r + 1 = 256$

$r = \frac{256}{3}$ (not valid) $r = \frac{255}{3} = 85$

$∴$ Coefficient of $x^{256}$ is $- (- 1)^{100} C_{85} =^{100} C_{15}$

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The coefficient of x256 in the expansion of (1−x)101(x2+x+1)100 is

The correct option is A 100C15(1−x)101(x2+x+1)100 ⇒(1−x)((1−x)(1+x+x2))100 ⇒(1−x)(1−x3)100 General term in (1−x3)100 is 100Cr(−x3)r= 100Cr(−x)3r ∴x256 occurs if 3r=256 or 3r+1=256 r=2563 (not valid) r=2553=85 ∴ Coefficient of x256 is −(−1)100C85= 100C15

The coefficient of $x^{256}$ in the expansion of $(1 - x)^{101} (x^{2} + x + 1)^{100}$ is