2014 AMC 12B 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\qquad \textbf{(B) }684\qquad \textbf{(C) }1024\qquad \textbf{(D) }1576\qquad \textbf{(E) }2016\qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.