Scenario: A computer system allows a user to choose a password with a length of one to eight characters. A system administrator needs to figure out how long a user's password should expire.

Assume that 1 million passwords can be tested per second by a hacker. The hacker can guess a user's password 24 hours a day and 7 days a week. A password should be expired if it has an equal or greater than 10% probability of having been guessed. Given each of the following condition, determine the time (in measurement of days) users' passwords should be expired.

Probability of having been guessed = total number of guess/total possible combinations.

You may Excel's Power function to calculate mⁿ. Make sure that you show your steps for partial credit.

1. Password characters may be any ASCII characters from 1 to 127, inclusive.

1. Password characters may be any alphanumeric characters ("A" through "Z," "a" through "z," and "0" through "9").

1. Password characters must be digits.