 In discrete geometry, you have discrete points, some of which are joined by lines. One version of discrete geometry is known as taxicab geometry, because it describes how taxicabs drive around cities. In particular, we can imagine the lines as being streets and the points as intersections and the taxicabs have to drive along the streets. In theory. Typically, we show this as a bunch of grid lines. The grid lines define the possible paths and points exist only at the intersection of grid lines. Now, in Euclidean geometry, there is a unique straight line between any two points, but what's a straight line? Remember, definitions are the whole of mathematics. All else is commentary, so we could define a straight line as the path of least distance between the two points. We also called this a geodesic. See, if we want to define straight lines in discrete geometry, we also need a way to define distance. So the simplest version of taxicab geometry, we use the taxicab metric, and this is simply the number of points you pass through as you go from A to B, including the terminal point B. So if I have a path like this, we pass through a total of 13 points, and so the path shown has length 13. It should be clear that no other path can have a length less than 13, so this is a path of least distance, and so we could call this a straight line, but in deference to the fact that this doesn't actually look like it's straight, we'll call it a geodesic. What's worth noting is that there are other paths of length 13 between the two points. For example, this one or this one, and since all of these paths have least distance, all of these are geodesics, and you might be a combinatorialist if the first question you ask is how many, and in this case, how many geodesics exist between the two points. Suppose the points have coordinates defined in the usual way. To go from one point to another in a path of least distance, we need to take a certain number of horizontal steps and a certain number of vertical steps. Now, the order in which we take the steps doesn't make a difference. We could do all of our horizontal steps first, then all of our vertical steps, or we could do all of our vertical steps, then all of our horizontal steps, or we could mix them up. All that matters is the number of horizontal and vertical steps, and so we can interpret this as a combinatorial problem. We need to take some total number of steps, of which a certain number are horizontal and the rest are vertical. We can do this in p-choose-q ways. We could also choose the vertical steps. The results are the same. So let's try to find the number of geodesics between the points a and b. So we might begin by noting that the path of least distance includes 13 steps, 8 of which are horizontal, so there are 13 choose 8 geodesics between the two points.