Advent calendar 2020

First, consider placing 5 tiles in a 1×9 grid. There is only one way to do this:

To get the number of ways of placing 5 tiles in a 1×10 grid, imagine adding an extra blank square to either the start or end of the grid or between two of the counters. There are 6 places this tile could be inserted leading to 6 arrangements of 5 tiles in a 1×10 grid.

For 5 tiles in a 2×10 grid, you can first pick the columns the tiles go in (as a tile being in a column means nothing can be placed the columns either side, the number of ways to pick columns is the same and the number of wats to arrange 5 tokens in a 1×10 grid). For each of these column choices, there are two locations for each tile (top or bottom). This leads to a total number of arrangements of 6×2⁵=192.