 In taxicab geometry, the number of geodesics from one point to another can be computed as n choose m, where n is the distance between the two as measured by the taxicab metric, and m is either the vertical or the horizontal displacement. But what if we were required to go through certain points? We can use the multiplication principle. So if we wanted to find the geodesics from A to B that also pass through P and Q, we can find the number of geodesics from A to P, the number of geodesics from P to Q, the number of geodesics from Q to B, and then multiply them. The shortest path from A to P will use four horizontal steps and three vertical steps for a total of seven steps, and since four of those need to be horizontal, so there are seven choose four geodesics from A to P. The geodesics from P to Q use two horizontal and one vertical step for a total of three steps. So there's a total of three steps, and this time for variety, let's choose one of them to be vertical. So there are three choose one geodesics. Finally, from Q to B requires two horizontal and one vertical step for a total of three steps. So there are three choose two geodesics. So using the multiplication principle, the number of geodesics from A to B that go through both P and Q is... Another possibility is that some paths might not exist. A useful strategy in mathematics, and when crossing the street, look both ways. So let's consider the paths that would use the excluded path. The only path that would use the excluded line would go through P and then on to Q. And we know there are seven choose four paths from A to P, and five choose four paths from Q to B. So there are the product paths from A to B that go through the excluded segment. Since there are 13 choose eight paths between A and B, then there are possible paths between A and B that avoid the excluded segment. And if there are several excluded connections, we can proceed in the same way using the inclusion-exclusion principle. We'll take all possible paths, subtract the ones that use one of the forbidden paths, add back those that use two, subtract those that use three, and so on. For example, if we have two excluded paths to use the inclusion-exclusion principle, we take all geodesics, subtract those that use RS and those that use PQ, and then add back those that use both RS and PQ. There are 13 choose eight geodesics between A and B. The geodesics that use RS are going to be those from A to R, and from S to B, to go from A to R, there's a total of three steps, one of which has to be horizontal. So there's three choose one geodesics. To go from S to B, there's a total of nine steps, six of which have to be horizontal, and so there's nine choose six geodesics. And so there will be, and we'll subtract that amount. The geodesics that use PQ are going to be all those from A to P, and all those from Q to B, and so we'll have, so we'll want to subtract those as well. Now we have to add back in the paths that use both, and those are, and we can multiply these together to find the number of geodesics that pass through both of the prohibited edges, and we'll add those back in to get the number of geodesics between A and B that do not use RS or PQ. And if we have more excluded paths, we can proceed in the same way. So here we've excluded a path between T and V, and so we note there are geodesics between A and B that use this excluded path. We'll subtract that, but then we'll need to add back in the paths that use two of the excluded paths. Note that no geodesics use both RS and TV, because that would require some backtracking. But there are geodesics that use TV and then PQ, and again there's no geodesics that use all three. So from our geodesics between A and B that don't use RS or PQ, we'll also subtract the geodesics that use TV, add back in the geodesics that use two of them, and subtract the geodesics that use three of them, but remember there aren't any. And that gives us the number of geodesics that don't use RS, PQ or TV.