[Math] Counting lines and triangles detemined by points in a plane, no two of which are collinear

Question

So I'm having trouble starting this problem:

*Ten points labeled $A, B,C, D, E, F, G, H, I, J$ are arranged in a plane
in such a way that no three lie on the same straight line.

a. How many straight lines are determined by the ten
points?

b. How many of these straight lines do not pass through
point $A$?

c. How many triangles have three of the ten points as vertices?

d. How many of these triangles do not have $A$ as a vertex?*

I'm not looking for the answers to the problem, but rather some guidance or insight on how to take on this problem. Or at the least, a starting point and I'll take it from there.

Thanks.

Answer 1

If you choose 2 points out of the 10, then one straight line is defined. If you choose 3 points out of the 10, then one triangle is defined.

How many ways to choose 2 or 3 points out of 10? Or out of 9?

Since no three line on the same straight line, two lines defined by different pairs of points must be different, and any three points must form a triangle with positive area.