Path Counting using Pascal's Triangle?

am having trouble with this problem.

A network of city streets forms square bloacks as shown in the diagram below.
http://img182.imageshack.us/my.php?imag ... oolqs6.jpg

Jeanine leaves the library and walks toward the pool at the same time as Miguel leaves the pools and walks toward the lbrary. Neither person follows a particular route, except that both are always moving toward their destination. What is the probability that they will meet if they both walk at the same rate?

In addition, how would I solve this for a 1 by 1 grid, 2 by 2 grid, 3 by 3 grid,etc.?

I know that you have to use Pascal's Triangle and I think that they would have to meet on their "4th" moves. The answer in the book is 35/128 but I don't know how to get this.

galactus said:

It seems to me there are \(\displaystyle 2^{7}\) ways to get from the pool to the library or vice versa.

Click to expand...

How in the world do you get that?
To go from library to pool we go down 4 blocks and to the east 4 blocks.
That is some combination DDDDEEEE (making constant progress).
That is \(\displaystyle \binom {8}{4} =70\) number of paths.

I reckon I was just misconstrued. I thought since we had two choices to make at each intersection, then there would be 2^7 ways to get from one corner to another. I could reason the 35 comes from them meeting in the middle since they travel at the same rate. But where in the world would the 128 come from then?. That's why I was thinking 2^7.