# 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

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.

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:

curly brackets{ }        round brackets( )     square brackets[ ]hash#

### How many possible 5

Assuming both lowercase and uppercase letters are allowed and repetition is also allowed, we have a total of 14 choices for each letter in the 5-letter password. Therefore, we can form 14^5 = 537,824 such passwords.

### How many 3 letter passwords can be made using the letters A through Z if?

There are 15,600 different 3-letter passwords, with no letters repeating, that can be made using the letters a through z. A 3-letter password, with no letters repeating, using the letters a through z, is simply a permutation of 3 letters taken from the alphabet, which has 26 letters.

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

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. (5) times (5) times (5) times (5) times (5) = 3125. There are 3125 possible passwords, if a password may have repeated characters.

### How many 4 letter passwords can be made using the letters A through Z?

Answer and Explanation: There are 456,976 different 4-letter passwords that can be made using the letters a through z.