9-6. Petya wants to place 99 coins in the cells of a $2 \times 100$ board so that no two coins are in cells that share a side, and no more than one coin is in any cell. How many ways are there to place the coins?
Answer: 396.

Solution. Note that there will be exactly 1 empty column. Then, to the left of it, there are exactly two ways to arrange the tiles, and to the right of it, there are also exactly two ways to arrange the tiles.
![](https://cdn.mathpix.com/cropped/2024_05_06_1d29d5cb430346a58eb5g-23.jpg?height=330&width=1078&top_left_y=221&top_left_x=494)

In total, there are $2 \cdot 2=4$ ways to arrange the tiles for a fixed empty column - each way on the left side of the picture must be combined with each way on the right side of the picture.

However, this reasoning does not work when the empty column is the farthest left or the farthest right - in those cases, there are not 4, but only 2 ways to arrange the tiles. Therefore, in total, there will be $98 \cdot 4 + 2 + 2 = 396$ ways.