I have a problem for the number of the shortest ways from (0,0,0) to (M,N,P).
May I ask the shortest way is (M+N+P), and the total number of the shortest ways is (M+N+P)! / M!N!P! or (M+N+P) C (M) * (N+P) C (N) * (P) C (P) ?
because a journey from (0,0,0) to (M,N,P) consists of lists of segments where each segment is the length of a single block either east or north or up.
So, (M+N+P) can be arranged in any order. However, each M*easts or N*norths or P*ups can be interchangeable each other.
Thus, I need to remove the duplication, so I need to divide (M+N+P)! by M!N!P!.
May I ask if I am correct?
Happy New Year!