How many ways can the 32 (or less) pieces of a chess board be arranged in? Insofar as 2 kings are always present and pawns at the opposite end of the board are promoted into other pieces.
How many of these arrangments could never occur?

It does not sound real hard to find the number of arrangements of pieces.

However, determining which ones could never occur is another matter.

Some arrangements could never exist in an obvious way, such as when both Kings are in check. Other arrangements are less obvious, such as a passed pawn when no pieces have been captured (well that is kind of obvious, but exactly where passed pawns can exist and how many promoted pawns could have occurred given a certain number of captured pieces begins to get tricky).

Of course, even of those positions that are attainable, many of them could only be the result of cooperation between opponents rather than competition (i.e. opponents deliberately trying to reach a position without regard to competition).