$\text{ The coefficient of }{\mathrm{x}}^{24}\text{ in the expansion of }{\left(1+{\mathrm{x}}^2\right)}^{12}\left(1+{\mathrm{x}}^{12}\right)\left(1+{\mathrm{x}}^{24}\right)\text{ is: }$

$\begin{array}{l}{\left(1+x^2\right)}^{12}\left(1+x^{12}\right)\left(1+x^{24}\right)={\left(1+x^2\right)}^{12}\left(1+x^{12}+x^{24}+x^{36}\right)\\ =\left(1+{\mathrm{x}}^{12}+{\mathrm{x}}^{24}+{\mathrm{x}}^{36}\right)\left({}^{12}\mathrm{C}_0+{}^{12}\mathrm{C}_1{\mathrm{x}}^2+{}^{12}\mathrm{C}_2{\mathrm{x}}^4+\dots +{}^{12}\mathrm{C}_6{\mathrm{x}}^{12}+\dots +{}^{12}\mathrm{C}_{12}{\mathrm{x}}^{24}\right)\\ \text{∴ Coefficient of }{\mathrm{x}}^{24}\text{ is }1.{}^{12}\mathrm{C}_{12}+1.{}^{12}\mathrm{C}_6+1{}^{12}\mathrm{C}_0=1+{}^{12}\mathrm{C}_6+1={}^{12}\mathrm{C}_6+2\end{array}$