Determine the maximum number of intersections for five different lines.

Question

Two different lines can intersect in at most one point. Three different lines can intersect in at most three points, and four different lines can intersect in at most six points

Answer 1

If n is the number of lines then the number of intersections is p=½n(n-1).

The reason is that each new line intersects the existing lines once only but the existing lines already intersect, so the number of those intersections has to be added to the set of new ones.

If we have P intersection points already from n lines then when we add the (n+1)th we have n new intersections to add so we have P+n intersections. We can write this as p[n+1]=p[n]+n.

When n=2, p=1 by the formula. That’s the base case.

Assume the formula is true for all n then p[n+1]=½n(n-1)+n=½n²-½n+n=½n²+½n=½(n+1)n=p[n+1]. That’s proof by induction.

When n=5, p₅=½(5×4)=10. Or since we know p₄=6, then p₅=6+4=10 because the 5th line intersects the existing 4 lines which already contain 6 intersections.