The value of 
 $\left(\begin{array}{c}30\\ 0\end{array}\right)\left(\begin{array}{c}30\\ 10\end{array}\right)-\left(\begin{array}{c}30\\ 1\end{array}\right)\left(\begin{array}{c}30\\ 11\end{array}\right)+\left(\begin{array}{c}30\\ 2\end{array}\right)\left(\begin{array}{c}30\\ 12\end{array}\right)-\mathrm{......}+\left(\begin{array}{c}30\\ 20\end{array}\right)\left(\begin{array}{c}30\\ 30\end{array}\right)=$
 

 $$\begin{gathered} {(1-x)}^{30}={}^{30}C_0{x}^0{-}^{30}{C}_1{x}^1{+}^{30}{C}_2{x}^2+\mathrm{........}+{(-1)}^{30}{C}_{30}{x}^{30} \\ {(x+1)}^{30}=30c_0{x}^{30}+30c_1{x}^{29}+\mathrm{.................}+30c_{30} \end{gathered}$$
 
Multiplying (1) and (2) and  equating the coefficient of  $x^{20}$ on both sides, we get required     sum is equal to coefficient of  $x^{20}$  in  ${(1-x^2)}^{30},$ which is given by ${}^{30}C_{10}.$