If you draw $\displaystyle m$ straight lines in the plane consisting of $\displaystyle x_{1}$ parallel in one direction, $\displaystyle x_{2}$ parallel in different direction ... and $\displaystyle x_{n}$ parallel in another direction and no three of the lines meeting at a point, show that the number of intersection points is $\displaystyle \sum_{i<j} x_{i}x_{j} = \frac{1}{2}\left[m^2 - \sum_{j = 1}^{k} x_{j}^2\right]$