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Theoretical maximum number of email addresses

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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Theoretical maximum number of email addresses

by adaw » Sun Apr 25, 2010 1:34 am
What is the current theoretical maximum number of email addresses that can exist, given the following:

The maximum length of an email address is 320 characters (255 + 64 + @), and:


the domain part (everything after the '@') must not exceed 255 characters

the local part (everything before the '@') must not exceed 64 characters

the local part can be any variation of:

* Uppercase and lowercase English letters (a-z, A-Z)

* Digits 0 to 9

* Characters ! # $ % & ' * + - / = ? ^ _ ` { | } ~

* Character . (dot, period, full stop) provided that it is not the first or last character, and provided also that it does not appear two or more times consecutively.

(For the purpose of this question I'm ignoring uncommon practices like quoted strings).

If you know the answer, please explain how!
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by cbenk121 » Sun May 09, 2010 12:57 pm
adaw wrote:What is the current theoretical maximum number of email addresses that can exist, given the following:

The maximum length of an email address is 320 characters (255 + 64 + @), and:


the domain part (everything after the '@') must not exceed 255 characters

the local part (everything before the '@') must not exceed 64 characters

the local part can be any variation of:

* Uppercase and lowercase English letters (a-z, A-Z)

* Digits 0 to 9

* Characters ! # $ % & ' * + - / = ? ^ _ ` { | } ~

* Character . (dot, period, full stop) provided that it is not the first or last character, and provided also that it does not appear two or more times consecutively.

(For the purpose of this question I'm ignoring uncommon practices like quoted strings).

If you know the answer, please explain how!
Let's start with a more simple example, and then you can scale up from there. Suppose that you're going to dinner. You can get beef or chicken, fries or salad, and a root beer or pop for drink. So how many possibilities exist?

A problem like this you break into two components: the number of decision points, and the number of possibilities at each point. Then you multiply everything together to get the number of possibilities.

So, in my simple example, you have 3 decision points (entre, side, and drink). At each point, you have 2 possibilities. So you take 2 x 2 x 2 = 8.

Now, scaling it up, you say that you have 320 decision points in an email address. One of those has one possibility: @. The rest have 26 (lowercase letters) + 26 (upper case letters) + 10 (digits) + 20 (oddball characters & period) = 82 possibilities.

So you take 82^319 to get your answer.

UPDATE: This only accounts for how many possibilities there are if every character is taken up (i.e., the email address is 320 characters long). So essentially you'd have to do a summation: 82^319 + 82^318 + 82^317 ... + 82^2 (as you'd have to have at least one character on left and right side of @, but that assumption wasn't in your stated problem)
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