AMC 12 2014b Problem 23

The number 2017 is prime. Let $S = \sum \limits_{k=0}^{62} \dbinom{2014}{k}$. What is the remainder when $S$ is divided by 2017?

$\textbf{(A) }32 \textbf{(B) }684 \textbf{(C) }1024 \textbf{(D) }1576 \textbf{(E) }2016$