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

Question

The coefficient of x256 in the expansion of (1-x)101(x2+x+1)100 is: (A)  100 C16 (B)  100 C15 (C)  100 C16 (D)  100 C15

Tardigrade Admin · Accepted Answer

(1-x)100 ⋅(x2+x+1)100 ⋅(1-x) =((1-x)(x2+x+1))100(1-x) =(13-x3)100(1-x) =(1-x3)100(1-x) =(1-x3)100-x(1-x3)100 Required coefficient (-1) ×(- 100 C85) = 100 C85= 100 C15

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

$^{100} C_{16}$

$^{100} C_{15}$

$^{100} C_{16}$

$^{100} C_{15}$

$(1 - x)^{100} \cdot (x^{2} + x + 1)^{100} \cdot (1 - x)$
$= ((1 - x) (x^{2} + x + 1))^{100} (1 - x)$
$= (1^{3} - x^{3})^{100} (1 - x)$
$= (1 - x^{3})^{100} (1 - x)$
$= (1 - x^{3})^{100} - x (1 - x^{3})^{100}$
Required coefficient $(- 1) \times (-^{100} C_{85})$
$=^{100} C_{85} =^{100} C_{15}$

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

100C16​

100C15​

100C16​

100C15​

(1−x)100⋅(x2+x+1)100⋅(1−x) =((1−x)(x2+x+1))100(1−x) =(13−x3)100(1−x) =(1−x3)100(1−x) =(1−x3)100−x(1−x3)100 Required coefficient (−1)×(−100C85​) =100C85​=100C15​

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

$^{100} C_{16}$

$^{100} C_{15}$

$^{100} C_{16}$

$^{100} C_{15}$

$(1 - x)^{100} \cdot (x^{2} + x + 1)^{100} \cdot (1 - x)$
$= ((1 - x) (x^{2} + x + 1))^{100} (1 - x)$
$= (1^{3} - x^{3})^{100} (1 - x)$
$= (1 - x^{3})^{100} (1 - x)$
$= (1 - x^{3})^{100} - x (1 - x^{3})^{100}$
Required coefficient $(- 1) \times (-^{100} C_{85})$
$=^{100} C_{85} =^{100} C_{15}$