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Combinatorics Difficulty 2.5 Junior Find the answer Canada

Figure 1 shows an arrangement of $3$ lines with $1$ intersection point, and Figure 2 shows
an arrangement of $3$ lines with
$3$ intersection points.

What is the maximum number of intersection points that can appear in
an arrangement of $4$ lines?

Pick one

Solutions — 2

Solution 1

Each of the $4$ lines can
intersect each of the other $3$ lines
at most once.

This might appear to create $4 $\times$ 3 =
12$ points of intersection, but each point of intersection is
counted twice – one for each of the $2$ lines.

Thus, the maximum number of intersection points is $\dfrac{4 \times 3}{2} = 6$ .

The diagram below demonstrates that $6$ intersection points are indeed
possible:

Solution 2

Eloise spent a total of $\$765$ and the mean price of each water
pump was $\$85$ .

Thus, Eloise purchased $\frac{\$765}{\$85}=9$ water pumps.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.