Determine the number of pathways that we can take to go from point A to point B, if we can only travel along the intersected points and can only go South or East.

find the number of paths that you could take to travel from A and B if you are only allowed to move to the right or down through the grid. You cannot travel a path if the path is not drawn in .

Consider all scenarios in which we have a city on a flat plane in which all roads are straight lines; the points at which two (or more) roads meet are called junctions; and we have two points A and B that are accessible from each other by road.In this question you have to determine whether the following statement is true or false.Among all possible routes between A and B, the shortest route goes through the fewest number of junctions.The following points may be helpful:Consider using a pen and paper to draw a few maps when thinking about this problem.The statement is about all possible scenarios, not just a particular city map that you may happen to draw. The statement is about all maps imaginable (with the above-mentioned properties). Obviously you cannot draw all possible maps, so you need to reason about this problem.The statement is true if you can convince yourself that it holds in all possible maps for all possible pairs of locations A and B. The statement is false if you can draw a map in which the statement does not hold.You can assume all distances are finite.Is the statement true or false?

A university is constructing a new campus and the developers want to connect all the facilities in the campus with a biking-walking path that will minimize the amount of concrete pathway that will have to be built.  The following network shows the possible connecting paths between the facilities: For this network, the distance (in 100 ft increments) between nodes is given in the diagram, and in the table below.  Branch Distance1-2 (A) 21-4 (B) 233-6 (C) 6The length of the pathways that will connect all the facilities with the minimum amount of pathway construction is: (Remember the units are 100ft.  So, for instance, the distance between node 1 and 3 is 600 ft.)

A boy lives at X and wants to go to School at Z. From his home X he has to first reach Y and then Y to Z. He may go X to Y by either 7 bus routes or 8 train routes. From there, he can either choose 5 bus routes or 6 train routes to reach Z. How many ways are there to go from X to Z?

In a coordinate system, A is the origin and B is the point (15, 8). One has to go from A to B, taking steps only rightwards or upwards. If each rightward step is 3 units long and each upward step is 2 units long, in how many ways can one go from A to B?