Counting & Binomials

Go and think about it...

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Two lines

Two straight lines in the plane can have one point of intersection (where they cross).

It is also possible for them to have no points of intersection. How?

If they are identical lines, we could say that there are infinitely many points of intersection. From now on, though, we’ll assume that no two lines are identical.

So for two straight lines, the number of points of intersection is $0$ or $1$.

Three lines

For three straight lines in the plane, the number of points of intersection is $0$, $1$, $2$ or $3$.

Can you find ways of drawing three straight lines to produce these numbers of intersections?

Four lines

For four straight lines in the plane, how many points of intersection can there be?

To ask it another way, can you find ways of drawing four straight lines with $0$ points of intersection? $1$ point? $2$ points? $3$ points? …

Five lines and more

What would happen if we had five straight lines? Or six straight lines? Or more?

Can you find any general rules for the possible numbers of intersections?

You might like to consider the following questions as you explore this problem:

• With $n$ lines, what is the maximum possible number of intersections?

• With $n$ lines, are there any numbers of intersections which are impossible to achieve?