\input amstex \sum_{k=1}^{1012} \binom{2014-k}{1010} = \sum_{k=1}^{1004}  \binom{2014-k}{1010}=  \sum_{k=0}^{1004}  \binom{2014-k}{1010} - \binom{2014}{2010}=\sum_{k=0}^{1004}  \binom{2014-k}{1004-k} - \binom{2014}{2010}=\binom{2015}{2010}-\binom{2014}{2010}.