an example of permutations and combinations

[omar18000dz]

I'm faced with the following problem, which I think I have solved, but I need to know if it belongs to a general class of problems incombinatorics and whether it is mentioned in books so I can refer to them.

Let's suppose we have N symbols we want to assemble in a certain order. Say we have the symbols A, B, C ,D ,E and F in the following order 'ABCDEF'. We want to know how many words formed by the same symbols (or alphabet, but each symbol has be used only once) will share no symbol at a given position with the reference word 'ABCDEF'. For instance 'BADCFE' is one of such words. In general, the number of such words is given (recursively) as:

 

C(N) = (N-1)* (C(N-1) + C(N-2)) with C(2) = 1 and C(1) = 0

 

If one is interested in the number of words sharing M symbols (say for M = 2, 'ABCDEF' and 'ACDEBF' share two symbols in the same positions or order):

 

C(M,N) = (N!/(N-M! * M!)) * C(N-M)

 

Have you please come across such a problem, does it have a name, or can I use a book to refere to it please ?