${}^{30}\mathrm{C}_0{}^{30}\mathrm{C}_{10}-{}^{30}\mathrm{C}_1{}^{30}\mathrm{C}_{11}+{}^{30}\mathrm{C}_2·{}^{30}\mathrm{C}_{12}\dots +{}^{30}\mathrm{C}_{20}{}^{30}\mathrm{C}_{30}=$

$\begin{array}{l}(1-\mathrm{x}{)}^{30}={\mathrm{C}}_0-{\mathrm{C}}_1\mathrm{x}+{\mathrm{C}}_2{\mathrm{x}}^2+\dots +{\mathrm{C}}_{30}{\mathrm{x}}^{30}\\ {\left(1+\frac{1}{\mathrm{x}}\right)}^{30}={\mathrm{C}}_0+{\mathrm{C}}_1\frac{1}{\mathrm{x}}+{\mathrm{C}}_2\frac{1}{{\mathrm{x}}^2}+\dots .+{\mathrm{C}}_{30}{\left(\frac{1}{\mathrm{x}}\right)}^{30}\text{ where }{\mathrm{C}}_{\mathrm{r}}={}^{30}\mathrm{C}_{\mathrm{r}}\\ Now (1-\mathrm{x}{)}^{30}{\left(1+\frac{1}{\mathrm{x}}\right)}^{30}=\left({\mathrm{C}}_0-{\mathrm{C}}_1\mathrm{x}+\dots +{\mathrm{C}}_{30}{\mathrm{x}}^{30}\right)\left({\mathrm{C}}_0+{\mathrm{C}}_1\frac{1}{\mathrm{x}}+\dots +{\mathrm{C}}_{30}\frac{1}{{\mathrm{x}}^{30}}\right)\\ {\mathrm{C}}_0{\mathrm{C}}_{10}-{\mathrm{C}}_1{\mathrm{C}}_{11}+{\mathrm{C}}_2{\mathrm{C}}_{12}\dots +{\mathrm{C}}_{20}·{\mathrm{C}}_{30}=\text{ Coefficient of }\frac{1}{{\mathrm{x}}^{10}}\text{ in }(1-\mathrm{x}{)}^{30}{\left(\frac{\mathrm{x}+1}{\mathrm{x}}\right)}^{30}\\ =\text{ Coefficient of }{\mathrm{x}}^{20}\text{ in }{\left(1-{\mathrm{x}}^2\right)}^{30}={}^{30}\mathrm{C}_{10}.\end{array}$