A computer hacker is trying to break into a computer system by guessing the syst

Question

A computer hacker is trying to break into a computer system by guessing the system administrator's password. The hacker can try 7 passwords a minute, but if he tries unsuccessfully for more than 9 minutes a day, he will get caught. If he limits his attempts to at most 9 minutes each day, he will not get caught. The hacker knows that the administrator's password is at least 4 and at most 11 characters long, does not contain any character that is not a number, an uppercase letter, or a lowercase letter (no special characters such as spaces, punctuation marks, dollar signs, etc.), and that the first character is a letter. Case matters, and characters can be repeated. 1. How many passwords are there that start with a letter, contain only numbers and upper and lowercase letters, and are exactly 4 characters long? 2. How many passwords are there that start with a letter, contain only numbers and upper and lowercase letters, and are exactly 11 characters long? 3. How many passwords are there that start with a letter, contain only numbers and upper and lowercase letters, and are at least 4 and at most 11 characters long? 4. What fraction of the passwords that start with a letter, contain only numbers and upper and lowercase letters, and are at least 4 and at most 11 characters long can the hacker try in 8 years without getting caught? (Assume that every year lasts 365.25 days.)

Explanation / Answer

There are 26 lowercase letters, 26 uppercase letters, and 10 digits, so each character can have one of

62=(26*2 +10) values, except the first character which can have only 52=2*26 values because we know it's a letter.

1) First character has to be a letter so 52 possible outcomes
2nd character can be numbers, or letters (lowercase/uppercase) so 62 possible outcomes

There are 52 * 62^3 possible passwords of length 4 =52*62*61*60 = 11799840

2) There are 52 * 62^10 possible passwords of length 11 =52*62*61*60*59*58*57*56*55*54*53

= 2.028856475 * e19

3) There are 52 * 62^3 possible passwords of length 4, 52 * 62^4 possible passwords of length 5 ..................,52 * 62^10 possible passwords of length 11 .i.e = password of atleast 4 and atmost 11 characters = (52*62^3) + (52*62^4) + (52*62^5) + (52*62^6) +(52*62^7) + (52*62^8) + (52*62^9) +(52*62^10)   

= 11799840 +696190560 +40379052480 +2.3016e12+ 1.2889e14+ 7.0889e15+3.8280e17 +2.0289e19

=2.0679e19

4) Given that the hacker can try 7*9 passwords per day (7 passwords/minute * 9 minutes/day), that's 63 * 365.25 =23010.75 passwords per year. That's 23010.75 *8 = 184086 passwords in 8 years.

The fraction of the passwords that the hacker can try in 8 years is

=184086 /2.0679e19 =8.9023e15

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