A Particular Domino Tiling
A grid with 3 rows and 52 columns is tiled with 78 identical \(2 \times 1\) dominoes. How many ways can this be done such that exactly two of the dominoes are vertical?
Details and Assumptions:
The dominoes will cover the entire board. They are not allowed to jut over the board, or overlap with each other.
Rotations and reflections (of the board) count as distinct ways.
Convention: rows are horizontal and columns are vertical.