Mathematics - Binomial Theorem Question with Solution | TestHub

$$\begin{array}{rl}& {\left[1-\mathrm{x}-{\mathrm{x}}^2(1-\mathrm{x})\right]}^6=(1-\mathrm{x}{)}^6{\left(1-{\mathrm{x}}^2\right)}^6\\ & =\left[{}^6{\mathrm{C}}_0-{}^6{\mathrm{C}}_1\mathrm{x}+{}^6{\mathrm{C}}_2{\mathrm{x}}^2-{}^6{\mathrm{C}}_3{\mathrm{x}}^3+{}^6{\mathrm{C}}_4{\mathrm{x}}^4-{}^6{\mathrm{C}}_5{\mathrm{x}}^5+{}^6{\mathrm{C}}_6{\mathrm{x}}^6\right]\times \left[{}^6{\mathrm{C}}_0-{}^6{\mathrm{C}}_1{\mathrm{x}}^2+{}^6{\mathrm{C}}_2{\mathrm{x}}^4-{}^6{\mathrm{C}}_3{\mathrm{x}}^6+\dots .\right]\end{array}$$ Coefficient of ${\mathrm{x}}^7={}^6{\mathrm{C}}_1{}^6{\mathrm{C}}_3-{}^6{\mathrm{C}}_3{}^6{\mathrm{C}}_2+{}^6{\mathrm{C}}_5{}^6{\mathrm{C}}_1=120-300+36=-144$