The number of ordered $5\text{ - tuple }(\mathrm{u},\mathrm{v},\mathrm{w},\mathrm{x},\mathrm{y})\text{ where }\mathrm{u},\mathrm{v},\mathrm{w},\mathrm{x},\mathrm{y}\in [1,11]$ which satisfy the inequality  is $2^{{\sin }^2\mathrm{u}+3{\cos }^2\mathrm{v}}\cdot 3^{{\sin }^2\mathrm{w}+{\cos }^2\mathrm{x}}\cdot 5^{{\cos }^2\mathrm{y}}\ge 720\text{ is }$

Given 
$$\begin{gathered} 2^{{\sin }^2\mathrm{u}+3{\cos }^2\mathrm{v}}\cdot 3^{{\sin }^2\mathrm{w}+{\cos }^2\mathrm{x}}\cdot 5^{{\cos }^2\mathrm{y}}\ge 720=2^4\cdot 3^2\cdot 5^1 \\ The equality holds when \\ {\sin }^2\mathrm{u}={\cos }^2\mathrm{v}={\sin }^2\mathrm{w}={\cos }^2\mathrm{x}={\cos }^2\mathrm{y}=1 \end{gathered}$$
$$\begin{gathered} \Rightarrow \therefore \mathrm{u},\mathrm{w}\in \left\{\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2},\frac{5\mathrm{\pi }}{2},\frac{7\mathrm{\pi }}{2}\right\} \\ \Rightarrow \mathrm{x},\mathrm{y},\mathrm{v}\in \{\mathrm{\pi },2\mathrm{\pi },3\mathrm{\pi }\} \\ \Rightarrow \therefore \text{ no of ordered }5-\text{ tuples } \\ =4^2\cdot 3^3=432 \end{gathered}$$