Bino-Combi-Turals

$\begin{aligned} S & = \sum_{k=0}^{20} (-1)^k {20 \choose 2k} \quad \quad \small \color{#3D99F6}{\text{Since }{20 \choose 2k} = 0 \text{ for }2k > 20} \\ & = \sum_{k=0}^{\color{#3D99F6}{10}} (-1)^k {20 \choose 2k} \\ & = {20 \choose 0} - {20 \choose 2} + {20 \choose 4} - {20 \choose 6} + ... + {20 \choose 20} \end{aligned}$

We note that:

$\begin{aligned} (1+i)^{20} & = {20 \choose 0} + {20 \choose 1}i - {20 \choose 2} - {20 \choose 3}i + {20 \choose 4} - ... + {20 \choose 20} \\ \implies S & = \Re \left \{(1+i)^{20} \right \} \\ & = \Re \left \{(2i)^{10} \right \} \\ & = \Re \left \{2^{10}(-1) \right \} \\ & = \boxed{-1024} \end{aligned}$