Question

A password is a string of ten characters, where each character is a lowercase letter, a digit, or one of the eight special characters !, @, #, $, %, &, (, and ). A password is called awesome, if it contains at least one digit or at least one special character. Determine the number of awesome passwords.

Answer 1

Given,

1. password has 10 characters.
2. Number of lowercase letters = 26
3. Number of Digits = 10
4. Number of special characters = 8

This problem is as example of Inclusion - Exclusion principle.

We have 10 positions. Each of those positions can be occupied by any of
the 44(26 + 10 + 8) choices available.

Totally we 44 * 44 * 44 * 44 * 44 *44 * 44 * 44 * 44 * 44 = 44^10.

There are 44^10 different passwords possible with given input.
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Among them is we remove the passwords with NO digit, we get passwords with
atleast one digit.

If we ignore digits, then total possible choices are 26 + 8 = 34.
We have 10 positions available.
So the passwords with NO digits are = 34^10.

Total passwords with atleast a digit = 44^10 - 34^10 = x (for easiness)
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Similarly, if we remove the passwords with NO characters, we get passwords with
atleast one character.

If we ignore characters, then total possible choices are 26 + 10 = 36.
We have 10 positions available.
So the passwords with NO character are = 36^10.

Total passwords with atleast a character = 44^10 - 36^10 = y (for easiness)
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Total number of passwords possible with atleast one character OR atleast
one digit = x + y.