In how many ways can one put 2 distinguishable objects on a 4x4 board?
In how many ways can one put them so that when you rotate the board to 90 degrees the positions of objects is preserved?
In how many ways can one put 2 distinguishable objects on a 4x4 board?
In how many ways can one put them so that when you rotate the board to 90 degrees the positions of objects is preserved?
There are $16$ different positions, so there are $16$ options for placing the first object and $15$ options for placing the second, for a total of $16\cdot15=240$.
There is no way to put them such that positions are preserved under rotations of $90^\circ$, even if they were indistinguishable, since no two positions on the board are transformed into each other under such rotations.