Question

What is the coefficient of x^8 in expansion of (2+ 3x+ 4x^3)^20 ?​

Answer 1

$(2+ 3x+ 4x^3)^{20} = \left(4x^3 + \left(2 + 3x\right ) \right )^{20} \\ \\ \sum\limits_{r=0}^{20} \underbrace{\binom{20}{r}}_{C_1} \cdot \underbrace{(4x^3)^{r}}_{C_2} \cdot \underbrace{(2 + 3x)^{20-r}}_{C_3}</p><p> \\ \\ \\ \text{To get K = coeff. of } x^8 : \\ </p><p> \\ \scriptsize{A_r = \binom{20}{r}; B_r = x^{3r}; m = 8 - 3r; C_r = \binom{20-r}{m} 2^{20-r-m} \cdot 3^m \cdot x^m} \\ \\ </p><p>\text{When r = 0 :} m = 8 - 0 = 8 \\ \\ </p><p> \small{A_0 = \binom{20}{0}; B_0 = 4^0 ; C_0 = \binom{20-0}{8} \cdot 2^{20-0-8} \cdot 3^8 } \\ \\ </p><p> \small{D_0 = A_0 \cdot B_0 \cdot C_0 = 3,385,299,640,320} \\ \\ </p><p>\text{When r = 1 :} m = 8 - 3 = 5 \\ \\ </p><p> \small{A_1 = \binom{20}{1}; B_1 = 4^1 ; C_1 = \binom{20-1}{5} \cdot 2^{20-1-5} \cdot 3^5} \\ \\ </p><p> \small{D_1 = A_1 \cdot B_1 \cdot C_1 = 3,703,575,674,880} \\ \\ </p><p>\text{When r = 2 :} m = 8 - 6 = 2 \\ \\ </p><p> \small{A_2 = \binom{20}{2}; B_2 = 4^2 ; C_2 = \binom{20-2}{2} \cdot 2^{20-2-2} \cdot 3^2} \\ \\ </p><p> \small{D_2 = A_2 \cdot B_2 \cdot C_2 = 274,338,938,880} \\ \\ </p><p>\text{Coeff. of }x^8 = D_0 + D_1 + D_2 = 7,363,214,254,080 \\ \\ \:$