Counting paths from the origin to a given point Consider the following "walk" from the origin (0,0) in the plane to the point E=(5,5). A walk consists of starting at the origin and on each move, moving either one unit distance up or one unit distance to the right. Each walk that starts at the origin and ends up at E=(5,5) traces out a continuous path consisting of horizontal and vertical line segments which starts at (0,0) and ends at (5,5). How many different such paths are there? 
(Hint: Note that to reach E=(5,5) from the origin, you have to make 10 moves total of which 5 have to be upward and 5 rightward.) 
I have absolutely no idea what this is asking. I asked my professor to clarify... has not responded yet...
 A: As the hint suggested, to reach $(5,5)$ from $(0,0)$, we will take $10$ consecutive "steps," of which $5$ will be up and $5$ to the right. We can choose any $5$ of these $10$ steps to be the "up" steps. 
So there are $\binom{10}{5}$ possible paths. 
A: To visualize the problem draw the quarter xy plane, that is, the positive x and y axises, on a graphing sheet. Now mark all the points with integer coordinates on and inside the square whose vertices are (0,0),(0,5),(5,0), and (5,5). You can draw small dots to make these marks. So, for example, you would have a dot on your graphing paper at the point (1,1). When you are done, you should have a total of 25 points. Now, the question asks, how many ways are there, starting from the origin (0,0), to move to the corner point (5,5), with the constraint that you can only move up or right and you can only step on the drawn points? A sample path would be (0,1)(0,2)(1,2)(2,2)(3,2)(3,3)(3,4)(4,4)(5,4)(5,5). Notice that you can't move diagonally to a point. Only up and right. Obviously you can count all the distinct paths by actually drawing them out or you can use combinatorics and counting methods to compute the number of paths. 
