Magic Non-Square

In how many ways can we fill a \(3 \times 5\) board with integers from \(1 \text{ to } 15\) such that the following conditions hold?

  1. Each integer is used exactly once.
  2. In all the rows, the sum of all the numbers in the row are equal.
  3. In all the columns, the sum of all the numbers in the column are equal.

Also:

  • All permutations of rows or columns count as distinct.
  • Transposing the board is not allowed.

Explicit example:

\[\begin{bmatrix} 1 & 3 & 11 & 12 & 13 \\ 8 & 7 & 9 & 10 & 6 \\ 15 & 14 & 4 & 2 & 5 \end{bmatrix}\]

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