Collinear points and straight lines and triangles

Question

There are 20 conplanar points of which 5 are collinear. How many straight line segments and how many triangles can be made using these points? I understand it belongs to combination. But there are both collinear and non collinear points. I don't know how to proceed.

Answer 1

No. of straight line segments: (if you think on sections): We can select any 2 from this 20 points. Thus the solution is $\binom{20}{2}=\frac{20!}{2!\cdot 18!}=20*19=\underline{\underline{380}}$

No. of triangles: selecting every 3 points were a valid triangle, if the aren't between the 5 collinear. Thus the solution is $\binom{20}{3}-\binom{5}{3}=20\cdot{19}\cdot{18}-5\cdot{4}\cdot{3}=\underline{\underline{6780}}$.

Answer 2

The number of lines are 20c2-5c2+1since 5 points are collinear and +1 is for the line connecting 5 collinear points Which is 181 . Now number of triangles is from 20 we can select 3 but 5 are collinear so we have to subtract them hence number of triangles is 20c3 -5c3=1130. So number of lines is 181 while number of triangles is 1130