9. To order books from an online site, the buyer must open an account. The buyer needs a username and a password.

a. If the username needs to be eight letters, how many different usernames are possible:

i. If the letters can be repeated?

ii. If the letters cannot be repeated?

b. If the password must be eight characters, which can be any of the 26 letters, 10 digits, and 12 special keyboard characters, how many passwords are possible:

i. If characters can be repeated?

ii. If characters cannot be repeated?

c. How would your answers to part (b) change if the password is case sensitive? (In other words, Password and password are considered different because the letter p is in uppercase

and lowercase.)

a)

i) 208827064576

ii) 62990928000

b)

i) 28179280429056

ii) 15214711438080

c)

i) 899194740203776

ii) 607681858331520

Step-by-step explanation:

a)

i) The alphabet consists of 26 letters.

For the first letter of the username there are 26 options.

For the second letter of the username there are 26 options.

For the third letter of the username there are 26 options.

For the fourth letter of the username there are 26 options.

For the fifth letter of the username there are 26 options.

For the sixth letter of the username there are 26 options.

For the seventh letter of the username there are 26 options.

For the eighth letter of the username there are 26 options.

To get the number of the possible usernames we multiply the number of options for each letter: 26 x 26 x 26 x 26 x 26 x 26 x 26 x 26 = 26^8=208827064576

ii) If letter cannot be repeated.

For the first letter of the username there are 26 options.

For the second letter of the username there are only 25 options.

For the third letter of the username there are only 24 options.

For the fourth letter of the username there are only 23 options.

For the fifth letter of the username there are only 22 options.

For the sixth letter of the username there are only 21 options.

For the seventh letter of the username there are only 20 options.

For the eighth letter of the username there are only 19 options.

To get the number of the possible usernames we multiply the number of options for each letter: 26 x 25 x 24 x 23 x 22 x 21 x 20 x 19 = 62990928000

b) 26 letters+10digits+12special characters=48 options

i) For the first letter of the password there are 48 options.

For the second letter of the password there are 48 options.

For the third letter of the password there are 48 options.

For the fourth letter of the password there are 48 options.

For the fifth letter of the password there are 48 options.

For the sixth letter of the password there are 48 options.

For the seventh letter of the password there are 48 options.

For the eighth letter of the password there are 48 options.

To get the number of the possible usernames we multiply the number of options for each letter: 48 x 48 x 48 x 48 x 48 x 48 x 48 x 48 = 48^8=28179280429056

ii) If letter cannot be repeated.

For the first letter of the password there are 48 options.

For the second letter of the password there are only 47 options.

For the third letter of the password there are only 46 options.

For the fourth letter of the password there are only 45 options.

For the fifth letter of the password there are only 44 options.

For the sixth letter of the password there are only 43 options.

For the seventh letter of the password there are only 42 options.

For the eighth letter of the password there are only 41 options.

To get the number of the possible usernames we multiply the number of options for each letter: 48 x 47 x 46 x 45 x 44 x 43 x 42 x 41 = 15214711438080

c) 26 uppercase letters+26 lowercase letters + 10digits+12special characters=74 options

i) For the first letter of the password there are 74 options.

For the second letter of the password there are 74 options.

For the third letter of the password there are 74 options.

For the fourth letter of the password there are 74 options.

For the fifth letter of the password there are 74 options.

For the sixth letter of the password there are 74 options.

For the seventh letter of the password there are 74 options.

For the eighth letter of the password there are 74 options.

To get the number of the possible usernames we multiply the number of options for each letter: 74 x 74 x 74 x 74 x 74 x 74 x 74 x 74 = 74^8=899194740203776

ii) If letter cannot be repeated.

For the first letter of the password there are 74 options.

For the second letter of the password there are only 73 options.

For the third letter of the password there are only 72 options.

For the fourth letter of the password there are only 71 options.

For the fifth letter of the password there are only 70 options.

For the sixth letter of the password there are only 69 options.

For the seventh letter of the password there are only 68 options.

For the eighth letter of the password there are only 67 options.

To get the number of the possible usernames we multiply the number of options for each letter: 74 x 73 x 72 x 71 x 70 x 69 x 68 x 67 = 607681858331520