Problem 5

Suppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maximum number of intersections that these perpendiculars can have.

Step 1 of 5: Count the perpendiculars

Previous step Next step

\binom51\binom42=30

Detailed analysis

Choose the point from which a perpendicular is drawn in $\binom51$ ways and the line joining two of the other four points in $\binom42$ ways. Hence there are $30$ perpendiculars.