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…
Best Answer
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.