Kings all over the game
2500 chess kings have to be placed on a \(100 \times 100\) chessboard so that
(i) no king can capture any other one (i.e., no 2 kings are placed in two squares sharing a common vertex);
(ii) each row and each column consists exactly 25 kings.
Find the number of such arrangements. (Two arrangements differing by rotation or symmetry are supposed to be different.)