What is the coefficient of x^11 in the expansion of (1+x²)⁴(1+x³)⁷(1+x⁴)¹² ??

Dear Student,

The expansion of x¹¹ in the expansion of ${\left(1+x^2\right)}^4{\left(1+x^3\right)}^7{\left(1+x^4\right)}^{12}$ can be found using binomial expansion
The x¹¹ can be achieved using following combination
$$\begin{gathered} x^2\times x^9+x^3\times x^8+x^8\times x^3+x^4\times x^4\times x^3 \\ {}^{4}C_1\times {}^{7}C_3+{}^{7}C_1\times {}^{12}C_2+{}^{4}C_4\times {}^{7}C_1+{}^{4}C_2\times {}^{7}C_1\times {}^{12}C_1 \\ \frac{4!}{3!}\times \frac{7!}{4!\times 3!}+7\times \frac{12!}{10!\times 2!}+1\times 7+6\times 7\times 12 \\ 4\times 35+7\times 66+7+504 \\ =1113 \end{gathered}$$

Answer= 1113

Regards,