Question 171
There are $4030$ numbers consisting of two of each number lying in between (and including) $1$ and $2015$. Suppose we line up these numbers from left to right. An ordered sequence of $2015$ numbers chosen from this line-up of $4030$ numbers and considered with the order inherited from the original line-up is called a *half-sequence*. What is the largest possible number of distinct half-sequences? Regard two half-sequences to be the same if they represent the same sequence of integers, even if they may come from different portions of the original line-up.