Join each of m points on the positive x-axis to each of n points on the positive y-axis. Assume that no three of the resulting segments are concurrent (except at an endpoint). How many points of intersection are there (excluding endpoints)?

**Solution**

Answer: mn(m-1)(n-1)/4.

Let us call the points on the y-axis y_{1} < y_{2} < ... < y_{n} and the points on the x-axis x_{1} < x_{2} < ... < x_{m}. Let us join first y_{1} to each x_{i}, then y_{2} and so on. Joining y_{1} to each of the x_{i} creates no intersections. Joining y_{2} to x_{1} creates m-1 intersections. Joining it to x_{2} creates m-2 intersections and so on. So, in all, joining y_{2} gives 1 + 2 + ... + m-1 = m(m-1)/2 intersections.

Similarly joining y_{3} to x_{1} gives 2(m-1) intersections. Joining it to x_{2} gives 2(m-2) and so on. So, in all, y_{3} gives 2 x m(m-1)/2. Similarly, y_{r+1} gives r x m(m-1)/2. So in total we get m(m-1)/2 (1 + 2 + ... + n-1) = mn(m-1)(n-1)/4.

© John Scholes

jscholes@kalva.demon.co.uk

15 Feb 2002