Question

five lines are drawn in the plane
(a) what is the smallest number of points of intersection of the five lines?
(b) what is the largest number of points of intersection?
(c) if m is an integer between the largest and smallest number of intersection points, can you arrange the lines to have m points of intersection? you are being asked to describe which arrangements work and which dont.

(a) the smallest number of intersections is

Explanation:

Step1: Analyze minimal intersections

To get the smallest number of intersection points, we can arrange all five lines to be parallel (or have all lines coincide, but parallel lines are a distinct case that also gives the same minimal count). Parallel lines never intersect each other.

Step2: Calculate minimal intersection count

If all 5 lines are parallel, there are 0 intersection points.

Step3: Analyze maximal intersections

For the largest number of intersections, every pair of lines must intersect at a unique point (no three lines meet at the same point). The number of unique pairs of lines is given by the combination formula $C(n,2) = \frac{n(n-1)}{2}$, where $n=5$.

Step4: Calculate maximal intersection count

Substitute $n=5$ into the combination formula:
$\frac{5\times(5-1)}{2} = \frac{5\times4}{2} = 10$

Step5: Analyze intermediate values

For any integer $m$ between 0 and 10:

• To get $m=1$: Have 4 parallel lines and 1 line crossing all 4 (this gives 4 intersections? No, correction: To get 1 intersection, have all 5 lines meet at a single common point.
• To get $m=2$: Have 3 parallel lines, and 2 other lines that intersect each other at one point, and each of these 2 lines intersects the 3 parallel lines. Wait, no: To get 2 intersections, we can have 4 lines meet at one point, and the 5th line intersects one of those 4 lines at a second unique point (and is parallel to the other 3? No, better: For any $m$ from 0 to 10, we can adjust the number of parallel lines and lines that intersect at unique points. For example:
• $m=0$: All parallel
• $m=1$: All lines concurrent (meet at 1 point)
• $m=2$: 3 lines concurrent, 2 lines parallel to each other and intersecting the concurrent lines at 2 unique points
• ...
• $m=10$: All lines in general position (no two parallel, no three concurrent)

Every integer between 0 and 10 is achievable by adjusting the arrangement of parallel and intersecting lines.

Answer:

(a) The smallest number of intersections is 0
(b) The largest number of intersections is 10
(c) Yes, every integer $m$ with $0 \leq m \leq 10$ is achievable. You can arrange lines by having sets of parallel lines and lines that intersect at unique points: for example, to get $m$ intersections, you can have $k$ parallel lines and the remaining $5-k$ lines arranged to intersect each other and the parallel lines at distinct points, adjusting $k$ and the concurrency of lines to reach the desired $m$.