(i) In this case, there are no restrictions on the password, so we can choose any 6 characters from the given set of 9 characters. This can be calculated using combinations. The number of different 6-character passwords that may be chosen is:
9C6 = 9! / (6!(9-6)!) = 84
(ii) In this case, the password must consist of 2 letters, 2 numbers, and 2 symbols, in that order. We can choose 2 letters from the given set of 3 letters, 2 numbers from the given set of 3 numbers, and 2 symbols from the given set of 3 symbols. This can be calculated using combinations. The number of different 6-character passwords that may be chosen is:
3C2 * 3C2 * 3C2 = (3! / (2!(3-2)!)) * (3! / (2!(3-2)!)) * (3! / (2!(3-2)!)) = 3 * 3 * 3 = 27
(iii) In this case, the password must start and finish with a symbol. We can choose 1 symbol from the given set of 3 symbols for the first and last positions. For the remaining 4 positions, we can choose any 4 characters from the remaining 6 characters (3 letters, 3 numbers, and 1 symbol). This can be calculated using combinations. The number of different 6-character passwords that may be chosen is:
3C1 * 6C4 = 3 * (6! / (4!(6-4)!)) = 3 * (6! / (4!2!)) = 3 * 15 = 45
Therefore, the number of different 6-character passwords that may be chosen is:
(i) 84
(ii) 27
(iii) 45