The coefficient of   $x^7$ in the expansion of ${\left(1-x-x^2+x^3\right)}^6$is

${\left(1-x-x^2+x^3\right)}^6=(1-x{)}^6{\left(1-x^2\right)}^6$

coefficient of $x^7$ in $\left[1{-}^6{C}_{1}x{+}^6{C}_2{x}^2-\cdots {+}^6{C}_6{x}^6\right]$

= coefficient of in $\left[1{-}^6{C}_{1}x{+}^6{C}_2{x}^2-\cdots {+}^6{C}_6{x}^6\right]\times \left[1{-}^6{C}_1{x}^2{+}^6{C}_2{x}^4-\cdots {+}^6{C}_6{x}^{12}\right]$

              

$\begin{array}{l}=\left({-}^6{C}_1\right)\left({-}^6{C}_3\right)+\left({-}^6{C}_3\right)\left({ }^6{C}_2\right)+\left({-}^6{C}_5\right)\left({-}^6{C}_1\right)\\ =-144\end{array}$