Total possibilities with no restrictions
12 steps altogether. 5 are horizontal 12C5=792 just as you got
Total number that go to (4,3)
7 steps to get from start to (4,3) four are horizontal = 7C4 = 35 ways
Number of ways from (4,3) to (5,7)
5 steps, 1 horizontal = 5 ways
35*5=175 ways to get from start to finish if you go through (4,3)
Number of ways to get from start to finish if you do not go through (4,3) is 792-175 = 617