Assume that the following figure represents a map of part of a city very well traced, where the lines are streets. A person is at point A and you want move to point B, but the condition is that you can only choose trajectories through the streets to the right or upwards. According to this condition, how many different trajectories exist to move from point A until point B?
I think that each vertex of each frame that makes up the figure generates two possible movements but I do not know what else to do. Could anyone help me, please?
This is a good observation but not quite true: since the movements are restricted to rightward and upward, once the path reaches a vertex directly below (or to the left of) B, the rest of the path is determined: if the path is directly below B, then no more rightward movement is possible, since this would necessitate future leftward movement to reach B.
In fact, the standard solution to this problem starts from a different observation: the path from A to B will always be a sequence of rightward and upward moves, so can be described as a word made up of the letters R and U. You can convince yourself that the word describing any path from A to B must have nine Rs and nine Us, and, conversely, any word with nine Rs and nine Us describes a unique path from A to B. The problem then becomes to compute the number of words one can make with nine Rs and nine Us; perhaps this is a type of problem you've seen before?