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Algebra Level 5

∑ k = 1 2015 ( k + 2016 ) ! ( k − 1 ) ! = ( A ) ! ( B ) ! ( C ) \sum _{k=1}^{2015}\frac{\left(k+2016\right)!}{\left(k-1\right)!}=\frac{\left(A\right)!}{\left(B\right)!\left(C\right)}

Find the value of A + B + C A+B+C


The answer is 8064.

Julian Poon by Julian Poon , 20, Singapore

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1 solution

Patrick Corn
Sep 22, 2017

The sum is 2017 ! ∑ k = 1 2015 ( k + 2016 2017 ) , 2017! \sum_{k=1}^{2015} \binom{k+2016}{2017}, which equals 2017 ! ( 4032 2018 ) = 4032 ! ( 2014 ! ) 2018 , 2017! \binom{4032}{2018} = \frac{4032!}{(2014!)2018}, so the answer is 4032 + 2014 + 2018 = 8064 . 4032+2014+2018 = \fbox{8064}.

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