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

=Coefficient of $x^{18}\text{ in }(1+x)(1-x{)}^{10}{\left(1+x+x^2\right)}^9$

= Coefficient of $x^{18}\text{ in }\left(1-x^2\right){\left[(1-x)\left(1+x+x^2\right)\right]}^9$

= Coefficient of  $x^{18}\text{ in }\left(1-x^2\right){\left(1-x^3\right)}^9$

=Coefficient of $x^{18}\text{ in }\left(1-x^2\right)\left(1{-}^9{C}_1{x}^3{+}^9{C}_2{x}^6\right.$${{\left.{-}^9{C}_3{x}^9+\dots {+}^9{C}_6{x}^{18}-\dots {-}^9{C}_9{x}^{27}\right)}^{2^x}}^9{C}_6=84$