9 |  |

In a square table of $9 \times 9$ cells, 9 cells are marked at the intersections of the second, fifth, and eighth rows with the second, fifth, and eighth columns. How many ways are there to get from the bottom-left cell to the top-right cell, moving only through unmarked cells upwards or to the right?
Let $a_{i, j}$ be the number of paths from the bottom-left cell to the cell ( $i, j$ ), not passing through the marked cells. Note that if the cell ( $i, j$ ) is not marked, then $a_{i, j}=a_{i-1, j}+a_{i, j-1}$ (and if it is marked, then $a_{i, j}=0$ ). Now it is not difficult to calculate these numbers for each cell of the board (see the figure).

| 1 | 1 | 8 | 39 | 39 | 114 | 339 | 339 | 678 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 |  | 7 | 31 |  | 75 | 225 |  | 339 |
| 1 | 2 | 7 | 24 | 41 | 75 | 150 | 225 | 339 |
| 1 | 1 | 5 | 17 | 17 | 34 | 75 | 75 | 114 |
| 1 |  | 4 | 12 |  | 17 | 41 |  | 39 |
| 1 | 2 | 4 | 8 | 12 | 17 | 24 | 31 | 39 |
| 1 | 1 | 2 | 4 | 4 | 5 | 7 | 7 | 8 |
| 1 |  | 1 | 2 |  | 1 | 2 |  | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

## Answer

678 paths.