Question

Find the coefficient of a⁴ in the product (1 + 2a)⁴ (2 − a)⁵ using binomial theorem.

Answer 1

$$\begin{gathered} (1+2a{)}^4(2-a{)}^5 \\ =[{}^{4}C_0(2a{)}^0+{}^{4}C_1(2a{)}^1+{}^{4}C_2(2a{)}^2+{}^{4}C_3(2a{)}^3+{}^{4}C_4(2a{)}^4]\times \\ [{}^{5}C_0(2{)}^5(-a{)}^0+{}^{5}C_1(2{)}^4(-a{)}^1+{}^{5}C_2(2{)}^3(-a{)}^2+{}^{5}C_3(2{)}^2(-a{)}^3+{}^{5}C_4(2{)}^1(-a{)}^4+{}^{5}C_5(2{)}^0(-a{)}^5] \\ =[1+8a+24a^2+32a^3+16a^4]\times [32-80a+80a^2-40a^3+10a^4-a^5] \\ \mathrm{Coefficient}\mathrm{of}{a}^4=10-320+1920-2560+512=-438 \end{gathered}$$