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Difference between revisions of "2014 AMC 12B Problems/Problem 23"
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==Problem==
 
==Problem==
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The number 2017 is prime.  Let <math>S = \sum \limits_{k=0}^{62} \dbinom{2014}{k}</math>.  What is the remainder when <math>S</math> is divided by 2017?
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The number <math>2017</math> is prime.  Let <math>S = \sum \limits_{k=0}^{62} \dbinom{2014}{k}</math>.  What is the remainder when <math>S</math> is divided by <math>2017?</math>
  
 
<math>\textbf{(A) }32\qquad
 
<math>\textbf{(A) }32\qquad
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== See also ==
 
== See also ==
 
{{AMC12 box|year=2014|ab=B|num-b=22|num-a=24}}
 
{{AMC12 box|year=2014|ab=B|num-b=22|num-a=24}}
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 +
[[Category:Intermediate Number Theory Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 13:22, 21 November 2020

Problem

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

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

Solution

Note that $2014\equiv -3 \mod2017$. We have for $k\ge1$ \[\dbinom{2014}{k}\equiv \frac{(-3)(-4)(-5)....(-2-k)}{k!}\mod 2017\] \[\equiv (-1)^k\dbinom{k+2}{k} \mod 2017\] \[\equiv (-1)^k\dbinom{k+2}{2} \mod 2017\] Therefore \[\sum \limits_{k=0}^{62} \dbinom{2014}{k}\equiv \sum \limits_{k=0}^{62}(-1)^k\dbinom{k+2}{2} \mod 2017\] This is simply an alternating series of triangular numbers that goes like this: $1-3+6-10+15-21....$ After finding the first few sums of the series, it becomes apparent that \[\sum \limits_{k=1}^{n}(-1)^k\dbinom{k+2}{2}\equiv -\left(\frac{n+1}{2} \right) \left(\frac{n+1}{2}+1 \right) \mod 2017 \textnormal{ if n is odd}\] and \[\sum \limits_{k=1}^{n}(-1)^k\dbinom{k+2}{2}\equiv \left(\frac{n}{2}+1 \right)^2 \mod 2017 \textnormal{ if n is even}\] Obviously, $62$ falls in the second category, so our desired value is \[\left(\frac{62}{2}+1 \right)^2 = 32^2 = \boxed{\textbf{(C)}\ 1024}\]

See also

2014 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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All AMC 12 Problems and Solutions

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