How many ways can I put down two indistinguishable pieces on an ordinary 8 times 8 chessboard, if the pieces must either be in the same row or be in the same column?

we have 448 ways of putting them.

Step-by-step explanation:

For the first piece we have no restrictions, so we have 8*8 = 64 ways of puting it. For the second piece we have 7 ways to put it in the same row and 7 ways of put it in the same column, so we have 14 ways.

This gives us a total of 14*64 = 896 ways.

However, since the pieces are indistinguishable, we need to divide the result by two, because we were counting each possibility twice, (if we swap the pieces, then it counts as the same way), thus we have 896/2 = 448 of putting two pieces on the board.