1. A password must be 6, 7, or 8 characters long, and may include capital
letters, lower-case letters, or digits. In how many ways could this be done
a. with no restriction?
26 capital letters + 26 lower case letters + 10 digits = 62 characters
62^6 + 62^7 + 62^8 = 221918520426688
b. with no repetition permitted?
62P6 + 62P7 + 62P8 = 138848807594160
c. if at least one of the characters in part a) must be a digit?
That will be the answer to (a) minus the number with no digits.
With no digits:
26 capital letters + 26 lower case letters = 52 characters
52^6 + 52^7 + 52^8 = 54507570843648
Subtracting from the result of part (a)
221918520426688 - 54507570843648 = 167410949583040
Edwin