Question

4 distinguishable chess pieces are to be placed on a chessboard such that no piece lies on a vertical or horizontal line with any other piece. In how many ways can the 4 pieces be placed?

Answer 1

`67,737,600`

Explanation:

I'm going to approach this problem in the following way:

First, let's draw a chessboard and label each square:

`((A1,A2,A3,A4,A5,A6,A7,A8),(B1,B2,B3,B4,B5,B6,B7,B8),(C1,C2,C3,C4,C5,C6,C7,C8),(D1,D2,D3,D4,D5,D6,D7,D8),(E1,E2,E3,E4,E5,E6,E7,E8),(F1,F2,F3,F4,F5,F6,F7,F8),(G1,G2,G3,G4,G5,G6,G7,G8),(H1,H2,H3,H4,H5,H6,H7,H8))`

Let's first notice that with an 8 x 8 chessboard, we can place the first piece on any of the 64 squares. For this purpose, let's put it in on A1.

This also means that all the A and 1 spaces are now unavailable, leaving 49 spaces left. Let's put our next piece on B2.

There are now 36 spaces left. Let's put the next piece on C3.

And now there's 25 spaces left - let's put a piece on D4.

Now let's realize that in all of these choices, we could have picked any of the 64, 49, 36, and 25 spaces. That gives us `64xx49xx36xx25=2,822,400` different pattern arrangements.

This is the answer if we had 4 indistinguishable pieces - say like 4 white pawns. But we have 4 distinguishable pieces - say the white pawn, queen, king, and rook. And so within the `2,822,400` different pattern arrangements, we have to also consider that each pattern arrangement can have `4!` ways to be done (for instance, with the pieces on A1, B2, C3, and D4, we can swap the pieces around in `4! =24` different ways. 1 pattern, 24 different arrangements of the pieces).

And so we have `2,822,400xx4! = 67,737,600` different ways to arrange the pieces.