$$\begin{aligned} & \text { General term }={ }^n C_r\left(7^{1 / 3}\right)^{n-r}\left(11^{1 / 12}\right)^r \\ & ={ }^n C_r(7)^{\frac{n-r}{3}}(11)^{r / 12} \end{aligned}$$

For integral terms, $r$ must be multiple of 12

$$\therefore \mathrm{r}=12 \mathrm{k}, \mathrm{k} \in \mathrm{~W}$$

Total values of $\mathrm{r}=183$

Hence $\max r=12(182)$

$$=2184$$

Min value of $n=2184$