The German Engima machine is a cryptographic device that creates a mapping from the alphabet to itself. This mapping is also based on previous key$-$presses: A could map to C originally but after more of a message is encrypted it could end up mapping to G$.$ The method to do this are the rotors. Each rotor consists of $26$ keys representing the letters of the alphabet. Each rotor forms a bijective mapping from one letter to another. Each rotor passed it$$s output to the next in the sequence, and there were three rotors selected from a pool of five. When a key was pressed, one rotor would rotate by one letter. Additionally, every $26$ keystrokes a second rotor would rotate as well. This meant that the machine was $26*26*25$ periodic. A reflector was used meaning that the encryption and decryption operations were equivalent. However, the implementation used meant that no letter could map to itself. There was also a plugboard that could map a letter to another before and after the rotor process $-$ up to $10$ pairs were be used.

How many different settings are possible assuming that $3$ of $5$ rotors were used, each rotor has $26$ positions, and $10$ plugboard pairs were always used?

How many different settings are possible assuming that $4$ of $5$ rotors were used, each rotor has $26$ positions, and $10$ plugboard pairs were always used?