Consider the following definitions for sets of characters:
• Digits = $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$
• Letters = $\{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z\}$
• Special characters = $\{*, \&, \$, \#\}$
Compute the number of passwords that satisfy the given constraints.
(i) Strings of length 7. Characters can be special characters, digits, or letters, with no repeated characters.
(ii) Strings of length 6. Characters can be special characters, digits, or letters, with no repeated characters. The first character can not be a special character.
How many different passwords are there that contain only digits and lower-case letters and satisfy the given restrictions?
(iii) Length is 7 and the password must contain at least one digit.
(iv) Length is 7 and the password must contain at least one digit and at least one letter.
