The coefficient of

Question:

The coefficient of $x^{18}$ in the product $(1+x)(1-x)^{10}\left(1+x+x^{2}\right)^{9}$ is :

  1. (1) 84

  2. (2) $-126$

  3. (3) $-84$

  4. (4) 126


Correct Option: , 2

Solution:

Given expression,

$(1-x)^{10}\left(1+x+x^{2}\right)^{9}(1+x)=\left(1-x^{3}\right)^{9}\left(1-x^{2}\right)$

$=\left(1-x^{3}\right)^{9}-x^{2}\left(1-x^{3}\right)^{9}$

$\Rightarrow$ Coefficient of $x^{18}$ in $\left(1-x^{3}\right)^{9}-$ coeff. of $x^{16}$ in $\left(1-x^{3}\right)^{9}$

$={ }^{9} C_{6}-0=\frac{9 !}{6 ! 3 !}=\frac{7 \times 8 \times 9}{6}=84$

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