Question

A computer password consists of six characters.

36a. 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? Please enter your result in scientific notation, making sure the answer in the left box is between 1 and 10.

36b. A computer system requires that passwords contain at least one digit. If six characters are generated at random, and each is equally likely to be any of the 26 letters or 10 digits, what is the probability that a valid password will be generated?

Answer 1

36a.

r =6 characters

Number of lower case letters(a to z) =26

Number of digits(0 to 9) =10

n =26+10 =36

Characters may be repeated. So, the total number of different passwords that are possible =n^r =36⁶ =2176782336

One character must be a digit. Number of different passwords that are possible using only lower case letters =26⁶ =308915776

Thus, the number of different passwords that are possible if each character may be any lowercase letter or digit, and at least one character must be a digit =36⁶ - 26⁶ =2176782336 - 308915776 =1867866560

In scientific notation: 1867866560 =1.86786656 x 10⁹

36b.

The probability that a valid password will be generated =number of valid passwords/total number of passwords =number of passwords with at least one character must be a digit/total number of passwords =(36⁶ - 26⁶)/36⁶ =1867866560/2176782336 =0.8581