Question 1200281
<br>
This problem could probably be solved using formal mathematics with combinatorics, by finding the total number of paths from A to B without the missing lines and subtracting the number of paths that would use either or both of the missing lines.<br>
But probably an easier method, since the diagram is relatively small, is the brute force method of counting the number of possible paths to each point on the grid.<br>
The diagrams below show the process, leading to the final answer.<br><pre>

   1 --- X --- X --- X --- X --- X --- X --- X --- X --- X
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- X --- X --- X --- X --- X     X --- X --- X --- X
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- X --- X --- X --- X --- X --- X --- X --- X --- X
   |     |           |     |     |     |     |     |     |
   |     |           |     |     |     |     |     |     |
   1 --- X --- X --- X --- X --- X --- X --- X --- X --- X
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1


   1 --- 5 --- X --- X --- X --- X --- X --- X --- X --- X
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 4 --- X --- X --- X --- X     X --- X --- X --- X
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 3 --- X --- X --- X --- X --- X --- X --- X --- X
   |     |           |     |     |     |     |     |     |
   |     |           |     |     |     |     |     |     |
   1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9 ---10
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1


   1 --- 5 ---12 --- X --- X --- X --- X --- X --- X --- X
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 4 --- 7 --- X --- X --- X     X --- X --- X --- X
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 3 --- 3 --- 7 ---12 ---18 ---25 ---33 ---42 ---52
   |     |           |     |     |     |     |     |     |
   |     |           |     |     |     |     |     |     |
   1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9 ---10
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1


   1 --- 5 ---12 --- X --- X --- X --- X --- X --- X --- X
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 4 --- 7 ---14 ---26 ---44    25 ---58 ---100 --152
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 3 --- 3 --- 7 ---12 ---18 ---25 ---33 ---42 ---52
   |     |           |     |     |     |     |     |     |
   |     |           |     |     |     |     |     |     |
   1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9 ---10
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1


   1 --- 5 ---12 ---26 ---52 ---96 ---121---179---279---431
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 4 --- 7 ---14 ---26 ---44    25 ---58 ---100 --152
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 3 --- 3 --- 7 ---12 ---18 ---25 ---33 ---42 ---52
   |     |           |     |     |     |     |     |     |
   |     |           |     |     |     |     |     |     |
   1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9 ---10
   |     |     |     |     |     |     |     |     |     |
   |     |     |     |     |     |     |     |     |     |
   1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1 --- 1</pre>
ANSWER: (If I didn't make any errors) 431<br>