Official solution
Given,
\[{\left( {1 - x - {x^2} + {x^3}} \right)^6}\]
= \[{\left[ {\left( {1 - x} \right) - {x^2}\left( {1 - x} \right)} \right]^6}\]
= \[{\left( {1 - x} \right)^6}{\left( {1 - {x^2}} \right)^6}\]
= \[\left( {1 + {}^6{C_1}( - x) + {}^6{C_2}{{( - x)}^2} + {}^6{C_3}{{( - x)}^3} + .......} \right)\times\]
\[\left( {1 + {}^6{C_1}( - {x^2}) + {}^6{C_2}{{( - {x^2})}^2} + {}^6{C_3}{{( - {x^2})}^3} + .......} \right)\]
\[\therefore\] Coefficient of x7 = \[ - {}^6{C_1} \times - {}^6{C_3} + \left( { - {}^6{C_3}} \right) \times {}^6{C_2} + \left( { - {}^6{C_5}} \right) \times - {}^6{C_1}\]
= 120 - 300 + 36 = - 144