SOLUTION: On a certain computer network each user has to have a password consisting of six to eight characters in length; each character is either a lowercase letter or a digit. A passwo

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Question 1054855: On a certain computer network each user has to have a password
consisting of six to eight characters in length; each character
is either a lowercase letter or a digit. A password must contain
at least one digit. How many possible passwords can be issued?
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

There are 26 lower case letters and 10 digits, 
that's 36 characters.

First of all we'll calculate the number whether they contain a 
digit or not, and then we'll subtract the ones with all letters.

Case 1: The number of passwords with 6 characters whether they contain
a digit or not.

Choose the 1st character any of 36 ways.
Choose the 2nd character any of 36 ways.
Choose the 3rd character any of 36 ways.
Choose the 4th character any of 36 ways.
Choose the 5th character any of 36 ways.
Choose the 6th character any of 36 ways.
That's (36)(36)(36)(36)(36)(36) = 366.

From that we must subtract the number of 6-character 
passwords with all letters.

Choose the 1st letter any of 26 ways.
Choose the 2nd letter any of 26 ways.
Choose the 3rd letter any of 26 ways.
Choose the 4th letter any of 26 ways.
Choose the 5th letter any of 26 ways.
Choose the 6th letter any of 26 ways.

That's (26)(26)(26)(26)(26)(26) = 266
That's 266 with no digits, just letters.

So the number of 6-character passwords is

 366 -  266 = 1867866560

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By the same reasoning the number of 7-character
passwords is 
 
 367 -  267 = 70332353920

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Also by the same reasoning, the number of 8-character
passwords is

 368 -  268 = 2612282842880

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Answer: 366 -  266 + 367 -  267 + 368 -  268 = 2684483063360

Edwin