In the expansion of ${\left(1+\mathrm{x}+\frac{7}{\mathrm{x}}\right)}^{11}$, the term not containing x is

${\left(1+\mathrm{x}+\frac{7}{\mathrm{x}}\right)}^{11}={\left(\frac{7+\mathrm{x}+{\mathrm{x}}^2}{\mathrm{x}}\right)}^{11}$

So, we have to find the coefficient of x¹¹ in ${\left(7+\mathrm{x}+{\mathrm{x}}^2\right)}^{11}$

Now $\begin{array}{l}{\left(7+\mathrm{x}+{\mathrm{x}}^2\right)}^{11}\end{array}$

       $\begin{array}{l}{\left.=\left[(7+\mathrm{x})+{\mathrm{x}}^2\right)\right]}^{11}\\ {=}^{11}{\mathrm{C}}_0(7+\mathrm{x}{)}^{11}{+}^{11}{\mathrm{C}}_1(7+\mathrm{x}{)}^{10}{\mathrm{x}}^2{+}^{11}{\mathrm{C}}_2(7+\mathrm{x}{)}^9{\mathrm{x}}^4\dots +\dots \end{array}$

$\begin{array}{rcl} & & \therefore \text{ Required coefficient }{=}^{11}{\mathrm{C}}_0{+}^{\mathrm{i}1}{\mathrm{C}}_1{\times }^{10}{\mathrm{C}}_9\times 7{+}^{11}{\mathrm{C}}_2{\times }^9{\mathrm{C}}_7\times 7^2{+}^{11}{\mathrm{C}}_3{\times }^8{\mathrm{C}}_5\times 7^3{+}^{11}{\mathrm{C}}_4{\times }^7{\mathrm{C}}_3\times 7^4{+}^{11}{\mathrm{C}}_5{\times }^6{\mathrm{C}}_1\times 7^5\\ & =& \sum _{\mathrm{r}=0}^5{ }^{11}{\mathrm{C}}_{\mathrm{r}}^{11-\mathrm{r}}{\mathrm{C}}_{11-2\mathrm{r}}{7}^{\mathrm{r}}\end{array}$