# Combinatorics Domain Names

I am working on the following problem and was wondering if people could check what I currently have as well as offer advice on how to do the last part of this problem:

"As of April $2006$, roughly $50$ million .com web domain names were registered (e.g., yahoo.com).

a. How many domain names consisting of just two letters in sequence can be formed? How many domain names of length two are there if digits, as well as letters, are permitted as characters? [Note: A character length of three or more is now mandated.]

b. How many domain names are there consisting of three letters in sequence? How many of this length are there if either letters or digits are permitted? [Note: All are currently taken.]

c. Answer the questions posed in (b) for four-character sequences.

d. As of April $2006$, $97,786$ of the four-character sequences using either letters or digits had not yet been claimed. If a four-character name is randomly selected, what is the probability that it is already owned?

What I was wondering was, say that I wanted to know the number of domains where lexicographic ordering was of concern; that is, if b was the first letter in the domain name, then a couldn't be the next one, only letters that follow b. Would I count in this manner, $25⋅24+24⋅23+23⋅22+22⋅21+...3⋅2+2⋅1=5152$? I'm sure there is an alternate way, too. Would it involve me finding the total number of domains, both in lexicographic order and not in lexicographic order, and subtracting something from that, right? Would $263=17576$ be the total number of combinations?"

I've got the following at the moment (please correct me if I'm wrong):

a. $\binom{26}2$ $\binom{36}2$

b. $\binom{26}3$ $\binom{36}3$

c. $\binom{26}4$ $\binom{36}4$

d. No idea.

For question a) only letters: $26^2$, because you have $26$ choices twice. If digits are allowed: $36^2$.
b) $26^3$ and $36^3$
d) There are $1679616$ four-character sequences. $\frac{97786}{1679616}=0.0582$. Thus, the probability of picking an occupied domain is roughly $6\%$.
For your additional question: The total number of two letter strings is $26^2$. The number of two letter strings with two equal characters is just $26$. The total number of two letter strings in lexicographic order is $\frac{26^2-26}2$, because exactly half of the strings with two different letters is in lexicographic order.
a little correction in (d) part, he is asking for prob. of picking already owned domain which will be $1-P(\mbox{non owned}) = (36^4 - 97786)/36^4 = .9417$