Problem 1. Let's consider the number of possibilities for a password. The following characters are allowed in our passwords: 26 lowercase letters ('a' to 'z'), 26 uppercase letters ('A' to 'Z'), and 10 digits ('0' to '1'). (a) How many possible password combinations are there of 8 to 12 characters? (This includes passwords of length 8 and 12.) (b) Suppose these passwords are no longer case sensitive. For example, 'a' and 'A' are no longer viewed as distinct characters. Now how many distinct possible password combinations are there? (c) Now, suppose our passwords are case sensitive again. Suppose passwords must be exactly 8 characters long and have exactly 1 number. How many password combinations are possible?

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