Question

A password is required to be 12 to 16 characters in length. Characters can be digits (0-9), upper or lower-case letters (A-Z, a-z) or special characters. There are 10 permitted special characters. There is an additional rule that not all characters can be letters (i.e. there has to be at least one digit or one special character.) How many permitted passwords are there? Give your answer in un-evaluated/un-simpli_ed form and explain it fully.

Answer 1

Answer:

The number of permitted number of passwords is $\sum\limits^{16}_{12} {^{72}P_{x}} - \sum\limits^{16}_{12} {^{52}P_{x}}$.

Step-by-step explanation:

A password is made of 12 - 16 characters.

Character options:

26 Upper case letters

26 Lower case letters

10 Numbers

10 Special characters

There are a total of 72 characters.

Let x = 12, 13, 14, 15 and 16.

The total number of passwords of length x is: $^{72}P_{x}$

The number of passwords formed with only letters is: $^{52}P_{x}$

Compute the total number of permitted passwords as follows:

Total no. of passwords = Total number of passwords of length x - Number of passwords formed with only letters of length x

                                      $=\sum\limits^{16}_{12} {^{72}P_{x}} - \sum\limits^{16}_{12} {^{52}P_{x}}$