Question

To determine the number of different passwords required for the computer.

Short answer

The number of possible different passwords required for the computer is \(9.9 \times {10^{21}}\).

Step-by-step solution

Given data

The password for computer must have at least 8 , but no more than 12 characters. Each character in the password is a lower case English letter, an uppercase English letter, a digit or one of the six special characters \(*, > , < ,!, + \) and =.

Concept used of number of ways to do a task

If a task is complete in\(n1\)ways or\(n2\) ways, then the number of ways to do the task is\(n1 + n2\).

Find the number of possible passwords

As per the given data,

Let us consider \({P_8},{P_9},{P_{10}},{P_{11}},{P_{12}}\) are the passwords having lengths \(8,9,10,11,12\)respectively.

Total number of possibilities for each character \(N = 2 \times 26 + 10 + 6 = 68\)

The number of possible passwords

\(\begin{array}{c}P = {P_8} + {P_9} + {P_{10}} + {P_{11}} + {P_{12}}\\ = {68^8} + {68^9} + {68^{10}} + {68^{11}} + {68^{12}}\\ = 9.9 \times {10^{21}}\end{array}\)

Therefore,

The number of possible different passwords required for the computer is \(9.9 \times {10^{21}}\)