How many 5 character passwords can be made using the letters A through Z?

EXAMPLE 1.5.4
The password for Gomer's e-mail account consists of 5 characters chosen
from the set {g, o, m, e, r} . How many arrangements are possible, if the password has no repeated characters?

SOLUTION

If the password contains no repeated characters, then forming a password involves nothing other than arranging the five characters of the set {g,o,m,e,r}. The number of ways to arrange 5 objects is 5 factorial
5 factorial = 120

There are 120 possible passwords.

How many 5-character passwords are possible if a password may have repeated characters?

SOLUTION

This is not a permutation (arrangement) problem, because it is possible to have repeated elements within one of these passwords. We can't use the permutation problem to solve this problem, so we will use the Fundamental Counting Principle.
In order to form a password, we need to make five decisions.

i. Choose first character: 5 options
ii. Choose second character: 5 options
iii. Choose third character: 5 options
iv. Choose fourth character: 5 options
v. Choose fifth character: 5 options
According to the Fundamental Counting Principle the number of outcomes is

(5) times (5) times (5) times (5) times (5) = 3125.

There are 3125 possible passwords, if a password may have repeated characters.

Questions and answers

There are some simple rules that you must follow when changing your password:

1. Your password must be at least 10 characters long.
2. Remember that UPPERCASE letters are different from lowercase letters (for example, A is treated as different from a).
3. It must contain at least one character that is not a letter, such as a digit.

The following special characters can be used in passwords changed using the My IT Account facility:

The following characters are also permitted:

• Uppercase [A-Z] and lowercase [a-z] English alphabet characters
• Digits 0-9
• spaces

The following are permitted, but may cause problems on some systems:

• 'greater than': >
• 'less than': <

You may not use any of the following characters:

• Any accented or non-english alphabetic characters: ü î ø å é etc.
• A leading space or trailing space

The upper limit for password length is set at 127 characters, however we recommend using something memorable.

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Categories

This is question number 839, which appears in the following categories:

The total number of sequences of 5 characters is $36^5$. Consider the sequences of 5 characters that are not valid passwords. There are two types of them: the sequences containing only digits ($10^5$ possible sequences) and the sequences containing only letters ($26^5$ possible sequences). This explains the (right) answer of the book.

Coming back to your answer, the problem is that you are counting this way the number of sequences having a digit in the first position and a letter in the second position.

Edit. If you want to use your method, you should decompose the cases as follows. Let $D$ denote a digit, $L$ a letter and $S$ any symbol. Then you may have the following mutually disjoint cases $$ LDSSS, LLDSS, LLLDS, LLLLD, DLSSS, DDLSS, DDDLS, DDDDL $$ leading to the number $$ 26 \times 10 \times 36^3 + 26^2 \times 10 \times 36^2 + 26^3 \times 10 \times 36 + 26^4 \times 10 + 10 \times 26 \times 36^3 + 10^2 \times26 \times 36^2 + 10^3 \times 26 \times 36 + 10^4 \times 26 = 48484800 $$ If you know automata theory, it is similar to finding the complement of a regular expression.

How many possible 5

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