Let $\sum _{r=0}^{20} a_r{x}^r={\left(1+x+x^2\right)}^{10}\text{ then }$$a_5{-}^{10}{C}_1\times a_4{+}^{10}{C}_2\times a_3{-}^{10}{C}_3\times a_2{+}^{10}{C}_4\times a_1{-}^{10}{C}_5\times a_0=$

 
multiply those two Equation , we get the given  series and is the coefficient    which is zero