You are supposed to create a 7-character long password for your mobile device.
(i) How many 7-character passwords can be formed from the 10 digits and 26 letters? (Only lowercase letters are taken throughout the problem.) Repeats are allowed, e.g., 0001a1a is a valid password.
(ii) How many of the passwords contain at least one of the 26 letters and at least one of the 10 digits? Write your answer in the form: (Answer to part i) $-$ (something).
(iii) How many of the passwords contain at least one of the 5 vowels, at least one of the 21 consonants and at least one of the 10 digits? Extend your method for part ii to write a formula and explain your reasoning.
(iv) Now suppose that in addition to the lowercase letters and digits, you can also use 12 special characters. How many 7-character passwords are there that contain at least one of the 5 vowels, at least one of the 21 consonants, at least one of the 10 digits and at least one of the 12 special characters? Write only the final formula analogous to your answer to part iii. Do NOT explain.