Binary Problem

From the given conditions, x must be a string of 2014 0s and 1s where the first digit is 1. There must be at least 1007 0s. Let the answer be s , then we can evaluate the answer by looking at how we can place the 0s in the 2013 free spaces. ie. $s = \displaystyle\sum_{i=1007}^{i=2013} \dbinom{2013}{i}$ Since $\dbinom{2013}{k} = \dbinom{2013}{2013-k}$ we get: $2s = \displaystyle\sum_{i=0}^{i=2013} \dbinom{2013}{i}$ . Using a well known result this gives us that $2s = 2^{2013}$ so $s = 2^{2012}$ which, when written in binary, is just a 1 followed by 2012 0s, so the sum of the digits is $\fbox{1}$