This is a combinatorics question:
Imagine you have an $8\times8$ empty chessboard.
You have $10$ identical pawns.
How many different ways can you place those $10$ pawns on the chessboard such that each pawn is at least (Euclidean) distance $X$ away from any of the other pawns?
This is just an example, but what I would like is a function where I can give it the dimensions of the chessboard, the number of pawns that must be on the chessboard, and the minimum distance each pawn must be from any of the other pawns, and then the function returns the number of possible arrangements.
If you don't know how to solve this problem, could you please let me know what subfield of combinatorics that I can learn to solve this problem? For example, is there a clever way of using inclusion-exclusion to solve this problem?
Lastly, if the above problem is too hard, how would you do this if instead of a chessboard that is $8\times 8$, you had a long board that was $64\times 1$?