Chapter 5.4, Problem 36E

Keep your password safe: A computer password consists of eight characters. Replications are allowed.

How many different passwords are possible if each character may be any lowercase letter or digit?

How many different passwords are possible if each character may be any lowercase letter?

How many different passwords are possible if each character may be any lowercase letter or digit, and at least one character must be a digit?

A computer is generating passwords. The computer generates eight characters at random, and each is equally to be any of the 26 letters or 10 digits.

Replications are allowed. is die probability that the password will contain all letters?

A computer system requires that passwords contain at least one digit. If eight characters are generated at random what is the probability that dry will form a valid password?

To determine

To find: the possible passwords having lowercase letter or digit

Answer to Problem 36E

The possible passwords having lowercase letter or digit are, $2.8211\times {10}^{12}$

Explanation of Solution

Given:

Computer password = 8 characters

(A to Z) 26 letters and (0 to 9)10 digits = 36 characters

Calculation:

The possible passwords having lowercase letter or digit are,

  $\begin{array}{c}\text{Number}\text{of}\text{ways}=\left\{\begin{array}{l}36\times 36\times 36\times 36\times \\ 36\times 36\times 36\times 36\end{array}\right\}\\ ={36}^8\\ =2.8211\times {10}^{12}\end{array}$

Conclusion:

Therefore, the possible passwords having lowercase letter or digit are, $2.8211\times {10}^{12}$

To determine

To find: the possible passwords having lowercase letter

Answer to Problem 36E

The possible passwords having lowercase letter are, $2.088\times {10}^{11}$

Explanation of Solution

Given:

Computer password = 8 characters

(A to Z) 26 letters and (0 to 9)10 digits = 36 characters

Calculation:

The possible passwords having lowercase letter are,

  $\begin{array}{c}\text{Number}\text{of}\text{ways}=\left\{\begin{array}{l}26\times 26\times 26\times 26\times \\ 26\times 26\times 26\times 26\end{array}\right\}\\ ={26}^8\\ =2.088\times {10}^{11}\end{array}$

Conclusion:

Therefore, the possible passwords having lowercase letterare, $2.088\times {10}^{11}$

To determine

To find: the possible passwords having lowercase letter or digit and should have at least one digit

Answer to Problem 36E

The possible passwords having lowercase letter or digit and should have at least one digit are, $2612282842880$

Explanation of Solution

Given:

Computer password = 8 characters

(A to Z) 26 letters and (0 to 9)10 digits = 36 characters

Calculation:

The possible passwords having lowercase letter or digit and should have at least one digit are,

  $\begin{array}{c}\text{Number}\text{of}\text{ways}=\left\{\begin{array}{l}36\times 36\times 36\times 36\times \\ 36\times 36\times 36\times 36\end{array}\right\}-\left\{\begin{array}{l}26\times 26\times 26\times 26\times \\ 26\times 26\times 26\times 26\end{array}\right\}\\ ={36}^8-{26}^8\\ =2821109907456-208827064576\\ =2612282842880\end{array}$

Conclusion:

Therefore, the possible passwords having lowercase letter or digit and should have at least one digit been, $2612282842880$

To determine

To find: the probability that password having all letters

Answer to Problem 36E

The probability that passwords having all letters are, $0.0740$

Explanation of Solution

Given:

Computer password = 8 characters

(A to Z) 26 letters and (0 to 9)10 digits = 36 characters

Calculation:

The possible passwords having all letters are, Let

E = the event of getting the password contains all letters.

  $\begin{array}{c}P\left(E\right)=\frac{\text{Favorable number of cases}}{\text{Total number of cases}}\\ =\frac{{26}^8}{{36}^8}\\ =\frac{208827064576}{2821109907456}\\ =0.0740\end{array}$

Conclusion:

The probability that passwords having all letters are, $0.0740$

To determine

To find: the possible passwords having at least one digit

Answer to Problem 36E

The possible passwords having at least one digit are, $0.09260$

Explanation of Solution

Given:

Computer password = 8 characters

(A to Z) 26 letters and (0 to 9)10 digits = 36 characters

Calculation:

The possible passwords having at least one digit are, Let

F = event of getting the password contains at least one digit.

  $\begin{array}{c}P\left(F\right)=\frac{\text{Favorable cases}}{\text{Total number of cases}}\\ =\frac{36-{26}^8}{{36}^8}\\ =\frac{2612282842880}{2821109907456}\\ =0.09260\end{array}$

Conclusion:

Therefore, the possible passwords having at least one digit are, $0.09260$