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? | 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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<cmath>\left(\frac{62}{2}+1 \right)^2 = 32^2 = \boxed{\textbf{(C)}\ 1024}</cmath> | <cmath>\left(\frac{62}{2}+1 \right)^2 = 32^2 = \boxed{\textbf{(C)}\ 1024}</cmath> | ||
| − | + | {{AMC12 box|year=2014|ab=B|num-b=22|num-a=24}} | |
| + | {{MAA Notice}} | ||
Revision as of 12:33, 21 February 2014
Problem
The number 2017 is prime. Let 
. What is the remainder when 
is divided by 2017?
Solution
Note that 
. We have for 
![\[\dbinom{2014}{k}\equiv \frac{(-3)(-4)(-5)....(-2-k)}{k!}\mod 2017\]](http://latex.artofproblemsolving.com/f/c/4/fc4df4114bc4b5853db7f0594b0bbd7bfe8016d5.png)
![\[\equiv (-1)^k\dbinom{k+2}{k} \mod 2017\]](http://latex.artofproblemsolving.com/7/2/9/729ef0f98878d86fed96a55dbb02a73ed0678d89.png)
![\[\equiv (-1)^k\dbinom{k+2}{2} \mod 2017\]](http://latex.artofproblemsolving.com/b/7/c/b7c09f87ee812d9d79beda60968eac083ee34880.png)
Therefore
![\[\sum \limits_{k=0}^{62} \dbinom{2014}{k}\equiv \sum \limits_{k=1}^{62}(-1)^k\dbinom{k+2}{2} \mod 2017\]](http://latex.artofproblemsolving.com/0/1/8/01872e9783dc697cc6c3f1895defbf6dc0417dc7.png)
This is simply an alternating series of triangular numbers that goes like this: 
After finding the first few sums of the series, it becomes apparent that
![\[\sum \limits_{k=1}^{n}(-1)^k\dbinom{k+2}{2}\equiv -\left(\frac{n+1}{2} \right) \left(\frac{n+1}{2}+1 \right) \mod 2017 \textnormal{ if n is odd}\]](http://latex.artofproblemsolving.com/d/7/5/d75b035918520665cb9d7cdc0d1331d161bb85c8.png)
and
![\[\sum \limits_{k=1}^{n}(-1)^k\dbinom{k+2}{2}\equiv \left(\frac{n}{2}+1 \right)^2 \mod 2017 \textnormal{ if n is even}\]](http://latex.artofproblemsolving.com/0/5/2/052ac2dbf84700f78612633b43c921b35cbd9a98.png)
Obviously, 
falls in the second category, so our desired value is
![\[\left(\frac{62}{2}+1 \right)^2 = 32^2 = \boxed{\textbf{(C)}\ 1024}\]](http://latex.artofproblemsolving.com/a/4/4/a44175c5e1f5de2bd10f00f3187f9ec2351a684b.png)
| 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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