9. Four different straight lines are drawn on a flat piece of paper. The number of points where two or more lines intersect is counted.
Which of the following could not be the number of such points?
A 1
B 2
C 3
D 4
E 5
SolUTION
B
Each of the figures below shows four straight lines drawn on a flat piece of paper with $1,3,4$ and 5 points, respectively, where two or more lines intersect.
It follows that these numbers of intersections are possible. We deduce from this that the correct option is B.
COMMENTARY
In the context of the SMC, when four of the given options have been eliminated, it is safe to conclude that the remaining option is correct. However, for a complete solution we need to give an argument to show that it is not possible to draw four straight lines so that there are exactly two points of intersection. We now give an argument to prove this.

We show that the attempt to draw four straight lines on a flat piece of paper so that there are just two intersection points is bound to fail.
Suppose that we aim to draw four straight lines so that there are only two points of intersection. Let these points be $P$ and $Q$.

For $P$ to be a point of intersection, there must be at least two lines through $P$. Therefore we need to draw at least one line, say $l$, through $P$ other than the line through $P$ and $Q$.
Similarly we need to draw at least one line, say $m$, through $Q$ other than the line through $P$ and $Q$.

The lines $l$ and $m$ must be parallel since otherwise there would be a third point of intersection.
We can now add the line through $P$ and $Q$, say $n$, without creating another intersection point. However, any other line through $P$ would not be parallel to $m$ and hence would create a third intersection point. So we cannot draw such a line. Similarly we cannot draw another line through $Q$ without creating a third intersection point.
So, having drawn the lines $l$ and $m$, the only line we can add without creating a third intersection point is $n$. It follows that we cannot draw four straight lines in such a way that there are exactly two intersection points.

FOR INVESTIGATION
9.1 Show that four straight lines can be drawn on a flat piece of paper with exactly six points of intersection.
9.2 Show that it is not possible to draw four straight lines on a flat piece of paper with more than six intersection points.
9.3 What are the possibilities for the number of intersection points when five straight lines are drawn on a flat piece of paper?
9.4 For $n \geq 6$, what are the possibilities for the number of intersection points when $n$ straight lines are drawn on a flat piece of paper?