.
Each pair of distinct point creates one and only one unique segment.
We do not consider pairs of coinciding points, since they create degenerated segments.
From the other side, each such a (non-degenerated) segment is created by a pair of distinct points from the given set.
Hence, there is ONE-TO-ONE correspondence between the segments under the question and the pairs of distinct points of the given 1001 points.
It implies that the number of segments under the question is equal to the number of pairs of distinct points, which is
= 
= 
= 1001*500 = 500500.
Answer. 500500 distinct line segments.
