Amy will walk south and east along the grid of streets shown. At the same time and at the same pace, Binh will walk north and west. The two people are walking in the same speed. What is the probability that they will meet?
I tried using Pascal's triangle but I have no idea how to proceed.
Best Answer
Here's one possible way that Amy and Binh can meet:
Together, the red and blue path make up a path from point $A$ to point $B$. There are $\binom{10}{5} = \frac{10!}{5!\,5!}$ such paths, because each path consists of $5$ vertical steps and $5$ horizontal steps, and there are $\binom{10}{5}$ ways to choose which steps are vertical.
There are $2^{5+5}$ ways to choose which $5$ steps Amy takes, and which $5$ steps Binh takes. Of them, $\binom{10}{5}$ result in them meeting. So the probability is $\binom{10}{5}/2^{10} = \frac{252}{1024} = \frac{63}{256}.$