Originally Posted by

**TheEmptySet** If you set up a grid with (0,0) in the bottom left corner( where you start) and (5,5) where your friend starts you should be able to convince yourself that you can only meet at the coordinates

(0,5),(1,4),(2,3),(3,2),(4,1),(5,0)

Now you just need to count the number of ways that you can get to each of these points.

Note the number of paths will be symmetric for points of the form (x,y) (y,x) i.e (0,5) will be the same of (5,0).

So there is one way to you and one way for you frend to get to (0,5)

There are $\displaystyle \binom{5}{1}$ ways for you and $\displaystyle binom{5}{1}$ for you friend to get to (1,4)

There are $\displaystyle \binom{5}{2}$ ways for you and $\displaystyle binom{5}{2}$ for you friend to get to (2,3)

Because of the symmetry we can add all of these up to get

$\displaystyle 2\left( \binom{5}{0}\binom{5}{0} + \binom{5}{1}\binom{5}{1} +\binom{5}{2}\binom{5}{2}\right)=2(1^2+5^2+10^2)=2 52$

There are $\displaystyle 2^10=1024$ paths total

for $\displaystyle \frac{252}{1024}$