$\left(\begin{array}{c}30\\ 0\end{array}\right)\left(\begin{array}{c}30\\ 10\end{array}\right)-\left(\begin{array}{c}30\\ 1\end{array}\right)\left(\begin{array}{c}30\\ 11\end{array}\right)+\dots \left(\begin{array}{c}30\\ 20\end{array}\right)\left(\begin{array}{c}30\\ 30\end{array}\right)$ is equal to

$$\begin{gathered} {}^{30}{\mathrm{C}}_0 {}^{30}{\mathrm{C}}_{10}-{}^{30}{\mathrm{C}}_1 {}^{30}{\mathrm{C}}_{11}+......{}^{30}{\mathrm{C}}_{20} {}^{30}{\mathrm{C}}_{30} \\ {(1+\mathrm{x})}^{30}={}^{30}{\mathrm{C}}_0+{}^{30}{\mathrm{C}}_1 \mathrm{x}+{}^{30}{\mathrm{C}}_2 {\mathrm{x}}^2+.........{}^{30}{\mathrm{C}}_{30} {\mathrm{x}}^{30} \\ {\left(1-\frac{1}{\mathrm{x}}\right)}^{30}={}^{30}{\mathrm{C}}_0-{}^{30}{\mathrm{C}}_1 \frac{1}{\mathrm{x}}+{}^{30}{\mathrm{C}}_2 \frac{1}{{\mathrm{x}}^2}+.......{}^{30}{\mathrm{C}}_{30} \frac{1}{{\mathrm{x}}^{30}} \\ {(1+\mathrm{x})}^{30} {\left(1-\frac{1}{\mathrm{x}}\right)}^{30}=\left({}^{30}{\mathrm{C}}_0+{}^{30}{\mathrm{C}}_1 \mathrm{x}+......+{}^{30}{\mathrm{C}}_{30} {\mathrm{x}}^{30}\right) \\ \left({}^{30}{\mathrm{C}}_0-{}^{30}{\mathrm{C}}_1 \frac{1}{\mathrm{x}}+......+{}^{30}{\mathrm{C}}_{10} \frac{1}{{\mathrm{x}}^{10}}-{}^{30}{\mathrm{C}}_{11} \frac{1}{{\mathrm{x}}^{11}}.....+{}^{30}{\mathrm{C}}_{30} \frac{1}{{\mathrm{x}}^{30}}\right) \end{gathered}$$

Comparing the Coeff of ${\mathrm{x}}^{-10}$ we get 

$$\begin{gathered} {}^{30}{\mathrm{C}}_0 {}^{30}{\mathrm{C}}_{10}-{}^{30}{\mathrm{C}}_1 {}^{30}{\mathrm{C}}_{11}+.......{}^{30}{\mathrm{C}}_{20} {}^{30}{\mathrm{C}}_{30} \\ =\frac{{({\mathrm{x}}^2-1)}^{30}}{{\mathrm{x}}^{30}}=\frac{{(1-{\mathrm{x}}^2)}^{30}}{{\mathrm{x}}^{30}}={}^{30}{\mathrm{C}}_{10} \end{gathered}$$