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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

Figure for this problem 2 lines. Thus, the maximum number of intersection points is

Figure for this problem $\text{4}$ 3}{2} = 6. The diagram below demonstrates that

Figure for this problem 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.

Figure for this problem

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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.