57. Suppose that a password for a computer system must have at least 8, but no more than 12, characters, where each character in the password is a lowercase English letter, an uppercase English letter, a digit, or one of the six spe- cial characters *, ,

Q: 57. Suppose that a password for a computer system must have at least 8, but no more than 12, characters, where each character in the password is a lowercase English letter, an uppercase English letter, a digit, or one of the six spe- cial characters *, >, <. !, +, and = a) How many different passwords are available for this computer system? b) How many of these passwords contain at least one OC- currence of at least one of the six special characters? c) Using your answer to part (a), determine how long it takes a hacker to try every possible password, assum- ing that it takes one nanosecond for a hacker to check each possible password.

For the math word problem about password combinations: ∻Key Concept∻ ⚹Combinatorics and Exponential Growth⚹ ∻Explanation∻ ⚹The number of different passwords can be calculated using combinatorics, considering the length of the password and the number of possible characters for each position.⚹ To solve part a, we need to calculate the total number of passwords for each length from 8 to 12 characters and then sum them up. Each character can be a lowercase letter (26 possibilities), an uppercase letter (26 possibilities), a digit (10 possibilities), or one of the six special characters, giving us a total of $26 + 26 + 10 + 6 = 68$ possible characters for each position. The number of different passwords of length $n$ is $68^n$. Therefore, the total number of different passwords is the sum of passwords of length 8 to 12: $$ \text{Total passwords} = 68^8 + 68^9 + 68^{10} + 68^{11} + 68^{12} $$ For part b, we calculate the number of passwords that do not contain any special character and subtract it from the total number of passwords. The number of such passwords of length $n$ is $62^n$ (since we only use letters and digits). $$ \text{Passwords without special characters} = 62^8 + 62^9 + 62^{10} + 62^{11} + 62^{12} $$ $$ \text{Passwords with at least one special character} = \text{Total passwords} - \text{Passwords without special characters} $$ For part c, we use the total number of passwords from part a and multiply by the time it takes to check one password (one nanosecond). $$ \text{Time to check all passwords} = \text{Total passwords} \times 1 \text{ nanosecond} $$ The actual calculations would require a computer or a calculator due to the large numbers involved.◊What algorithms could be used to efficiently calculate the number of different passwords available in this scenario?⍭ Generate me a similar question◊

Question

57. Suppose that a password for a computer system must have
at least 8, but no more than 12, characters, where each
character in the password is a lowercase English letter,
an uppercase English letter, a digit, or one of the six spe-
cial characters *, >, <. !, +, and =
a) How many different passwords are available for this
computer system?
b) How many of these passwords contain at least one OC-
currence of at least one of the six special characters?
c) Using your answer to part (a), determine how long it
takes a hacker to try every possible password, assum-
ing that it takes one nanosecond for a hacker to check
each possible password.

Accepted Answer

For the math word problem about password combinations: ∻Key Concept∻ ⚹Combinatorics and Exponential Growth⚹ ∻Explanation∻ ⚹The number of different passwords can be calculated using combinatorics, considering the length of the password and the number of possible characters for each position.⚹ To solve part a, we need to calculate the total number of passwords for each length from 8 to 12 characters and then sum them up. Each character can be a lowercase letter (26 possibilities), an uppercase letter (26 possibilities), a digit (10 possibilities), or one of the six special characters, giving us a total of $26 + 26 + 10 + 6 = 68$ possible characters for each position. The number of different passwords of length $n$ is $68^n$. Therefore, the total number of different passwords is the sum of passwords of length 8 to 12: $$ ext{Total passwords} = 68^8 + 68^9 + 68^{10} + 68^{11} + 68^{12} $$ For part b, we calculate the number of passwords that do not contain any special character and subtract it from the total number of passwords. The number of such passwords of length $n$ is $62^n$ (since we only use letters and digits). $$ ext{Passwords without special characters} = 62^8 + 62^9 + 62^{10} + 62^{11} + 62^{12} $$ $$ ext{Passwords with at least one special character} = ext{Total passwords} - ext{Passwords without special characters} $$ For part c, we use the total number of passwords from part a and multiply by the time it takes to check one password (one nanosecond). $$ ext{Time to check all passwords} = ext{Total passwords} imes 1 ext{ nanosecond} $$ The actual calculations would require a computer or a calculator due to the large numbers involved.◊What algorithms could be used to efficiently calculate the number of different passwords available in this scenario?⍭ Generate me a similar question◊

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