Three points that are not on a line determine three lines. How many lines are determined by seven points, no three of which are on a line?
I'm sure there's a more analytical, theoretical way to proceed, but here's a mental experiment I did to come up with the answer for the 7 point case:
Draw 3 points at the corners of a nice, equilateral triangle. You easily satisfy yourself that they determine 3 lines to connect the 3 points.
So we can say there's a function, f, such that f(3) = 3
Add a 4th point. Draw lines to connect all three prior points. You need 3 more lines to do this, for a total of 6.
f(4) = 6.
Add a 5th point. connect to all 4 prior points. You need 4 additional lines to do this, for a total of 10.
You start to see a pattern:
f(n) = f(n-1) + n-1
from this you can step through to the answer:
f(5) = f(4) + 4 = 10
f(6) = f(5) + 5 = 15
f(7) = f(6) + 6 = 21