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Binomial Coefficients modulo p

Source: 2014 AMC 12B #23

February 20, 2014
modular arithmeticinductionbinomial coefficientsPascal's TriangleAMC

Problem Statement

The number 20172017 is prime. Let S=∑k=062(2014k)S=\sum_{k=0}^{62}\binom{2014}{k}. What is the remainder when SS is divided by 20172017?
<spanclass=′latex−bold′>(A)</span>32<spanclass=′latex−bold′>(B)</span>684<spanclass=′latex−bold′>(C)</span>1024<spanclass=′latex−bold′>(D)</span>1576<spanclass=′latex−bold′>(E)</span>2016<span class='latex-bold'>(A) </span>32\qquad <span class='latex-bold'>(B) </span>684\qquad <span class='latex-bold'>(C) </span>1024\qquad <span class='latex-bold'>(D) </span>1576\qquad <span class='latex-bold'>(E) </span>2016\qquad
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