Arrangement of Counters

We place the counters in a row by row basis, so we need not to worry about placing them on the same row.

On the first row, there are 8 columns, hence 8 possibilities. On the second row, we have 7 available columns left because we cannot place it on the column that contains the first counter. The pattern continues, the total number of configurations are

$8\times 7 \times 6 \times \dots \times 1 = 8!=40320$

There are eight counters. Let's put it piece by piece on the board. The first piece can be placed anywhere on the board; thus we have $8^2=64$ ways to do that. Now for the second counter, it can't be on the same row and column of the first counter, so we have $7^2=49$ ways to do that. The third counter can't be on the same row and column of the first and second counter, so we have $6^2=36$ ways to do that and so on.

total number of ways = $\frac{(64)(49)(36)(25)(16)(9)(4)(1)}{8!}=40320$

Note that we divided it by $8!$ because the order in which the counters are placed is not important.