A grid with 3 rows and 52 columns is tiled with 78 identical 2*1 dominoes. How many ways can this be done in such that exactly 2 dominoes are vertical.
The dominoes will cover the entire board. They are NOT allowed to jut over the board, or overlap. Rotations count as distinct ways.
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Both the vertical dominoes have to appear in the same pair of rows - that is, both the vertical dominoes should have their ends in first and second row, or second and third row. : Otherwise, the top and bottom rows will have 51 cells to be filled by horizontal dominoes, which is not possible.
Also, we need to ensure that the three pieces the vertical dominoes divide the rows into should all have even length so that they can be filled by horizontal dominoes.
Number of ways of choosing the rows to place the vertical dominoes = 2
Number of ways of placing two vertical dominoes such that all three pieces of the row are even = number of solutions of the equation "a + b + c = 25". This is equal to 27 choose 2 = 351
Thus the required answer = 2 * 351 = 702