: Sum of the series $S=3^{-1}\left[\left({ }^{10}{C}_0\right)-3^{2 }\left({ }^{10}{C}_1\right)+3^2\left({ }^{10}{C}_2\right)-3^3\left({ }^{10}{C}_3\right)+....\dots +3^{10}\left({ }^{10}{C}_{10}\right)\right]$ is 

We have 

$\begin{array}{l}S=3^{-1}\left[{ }^{10}{C}_0{-}^{10}{C}_1\left(3\right){+}^{10}{C}_2\left(3^2\right){-}^{10}{C}_3\left(3^3\right)\right.\left.+\dots .....{+}^{10}{C}_{10}\left(3^{10}\right)\right]\\ \\ =\frac{1}{3}(1-3{)}^{10}=\frac{1}{3}\left(2^{10}\right)\end{array}$