Try a Simpler Version #1

How many line segments can be drawn through each pair of 15 points? No three points are in a straight line.

For today's problem, you could immediately try and draw the complete diagram:
Clearly, it will be too hard to count all the line segments in this diagram!
Could we sneak up on the problem by trying simpler versions of it? Fewer points perhaps?
What happens if we take only three or four points?
In these small cases, the number of segments is easy to count.
Now, how do you get from here back to the original problem with 15 points?

As always, be sure to explain the work you did to arrive at your answers!