Coefficient of $x^{11}$ in the expansion of ${\left(1+x^2\right)}^4{\left(1+x^3\right)}^7{\left(1+x^4\right)}^{12}$ is

We have 

$\begin{array}{l}{\left(1+x^2\right)}^4{\left(1+x^3\right)}^7{\left(1+x^4\right)}^{12}\\ ={\left(1+x^2\right)}^4\left(1+7x^3+31x^6\right.\\ \left.+35x^9+\dots \right)\left(1+12x^4+66x^8+\dots \right)\end{array}$

As expansion of ${(1 + x^2)}^4$ contains only even power, while multiplying last two expansions, we shall only consider odd powers.

$\begin{array}{l}\therefore {\left(1+x^2\right)}^4(1+{\left.x^3\right)}^7{\left(1+x^4\right)}^{12}\\ =\left(1+4x^2+6x^4+4x^6+x^8\right)\left(7x^3+84x^7\right.\\ \left.+35x^9+462x^{11}+\dots \right)\end{array}$

Thus, coefficient of $x^{11}$ is

$\text{ (1) }\left(462\right)+4\left(35\right)+\left(6\right)\left(84\right)+\left(1\right)\left(7\right)=1113$