Verified answer

First compute total paths from (0,0) to (4,4).

Each path consists of 8 steps, 4 vertical, 4 horizontal,

and they can be in any order, so they number 8c4 = 70

You arrange 4 x's and 4 y's in a line, representing whether you

go across or up at that point, and there are 8c4 ways to *choose*

where the x's go, leaving the other places for the y's.

Then subtract the ones that pass through (2,2)

there are 4c2 = 6 ways to get TO (2,2)

and another 4c2 = 6 ways to get FROM there to (4,4)

A total of 6 * 6 = 36 paths through (2,2)

Answer is 70 - 36 = 34.