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A company‘s computer system requires passwords  to be 7 to 9 characters long, and ...

A company‘s computer system requires passwords
 to be 7 to 9 characters long, and
 to consist of uppercase and lowercase letters {A, a, B, b, C, c, . . . , Z, z}, digits {0, 1, 2, 3, . . . , 9} and the special
characters {&, #, $, *, +, _}, and
 to contain at least one special character or digit.
How many such passwords are there?

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Answer #1

Total special characters = 6

Total digits = 10

Total alphabets = 26 + 26 = 52

Total characters = 6 + 10 + 52 = 68

Total possible passwords = 68^7 + 68^8 + 68^9

Password with no special character or digit = 52^7 + 52^8 + 52^9

Required passwords as per question = 68^7 + 68^8 + 68^9 - ( 52^7 + 52^8 + 52^9) = 2.8716592841031680 * 10^16

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