Let

Rn be the number of ways to place n non-attacking rooks on an nxn chess-board.

(a) Prove that |Rn|= n!.

(b) Define f : Rn =>Z by

f

(r) = number of rooks in r on the diagonal squares

(for r is an element ofRn). Is f injective? Is it surjective? Is there a partition of Rn described by f?

The first part i get that |Rn| is n^2!/n! because there are n^2 places to go and n rooks, not sure if right.

part b has me confused. It should be injective and surjective since there is only one choice for the rook to go to at one time. Any help would be great.