Question

What is the maximum number of intersections that can be formed by n lines?

Answer 1

look at the pattern

1 line --- 0 intersection
2 ines --- 1 intersection
3 lines ---- 3 intersections
4 lines ---- 6 intersections
5 lines --- 10 intersections
6 lines ---- 15 intersections
.....

the numbers of the sequence
1 3 6 10 15 .... are called the "triangular numbers"

(think of the sum of billiard balls as you rack up the balls using the triangles common to billiards.
Add up the balls as you add new rows)

for n lines it would be n(n-1)/2
e.g. for 10 lines it would be 10(9)/2 or 45