Suppose a certain web site has you choose a username and a password where the username consists of 9 alphanumeric characters and the password is 8 alphanumeric characters the last of which must be numeric and the first of which must be alphabetical.

a. How many usernames are possible if they are not case sensitive?

b. How many passwords are possible if they are not case sensitive?

c. How many usernames are possible if they are case sensitive?

d. How many passwords are possible if they are case sensitive?

Let us suppose that, when we say alphanumeric, it means that it is either a letter or a number (not any special characters such as #$% and so on is allowed). We are going to used the following counting principle which is know as the multiplication principle. It will be illustrated from an example. We have 5 different colors and we want to pick a color at two different moments. We can pick any of the 5 colors at anytime. Then if we want to know the total of different combinations, we must see how many positibilities we have when we want to choose. At the first moment, we have 5 colors available, and at the second moment we have also 5 colors available, therefore, the total number of different outcomes is 5 times 5, 25.

Consider that the usernames and passwords are considering only the english alphabet, so it has 26 letters. We will also consider the digits from 0 to 9, so this means we have 10 extra options.

a) If they are not case sensitive, we are going to consider that we have the lower case letters available, so for each character of the username, we have 26 letters and 10 digits avaialbe for a total of 36 options. Since we must choose 9 characters, we will have 36^9 different outcomes (36 times 36 a total of 9 times).

b) In the case of the passwords, we need to pay attention to the restriction. The first character must be a letter, so we have 26 options available, for the characters 2-7 we can have either a letter or a number, so we have 36 options, and for the last character we must have a number, so we only have 10 options. Therefore the total number of passwords are given as follows.

26 (number of letters for the first character) * (36^6) (36 per each of the characters 2-7) * 10 (for the last digit, which must be a number).

c) NOte that if we consider case sentivity, then we have that for each letter, say a, its upper case version is another letter, say A. So, we'll have the double of posibilities. In this case, we'll have 52 letters to choose and 10 numbers to choose to a total of 62 options.

Then, we will have 62^9 different usernames.

d) With the same idea, the number of passwords are given by

(52)*(62^6)*(10).