Question:
The coefficient of $x^{18}$ in the product $(1+x)(1-x)^{10}\left(1+x+x^{2}\right)^{9}$ is :
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 $\left(1-x^{3}\right)^{9}$
$={ }^{9} C_{6}-0=\frac{9 !}{6 ! 3 !}=\frac{7 \times 8 \times 9}{6}=84$