calculate the number of possible number of words If one word can be at most 63 characters long. It can be combination of :


*

*letters from a to z

*numbers from 0 to 9

*hyphen - but only if not in the first or the last character of the word


I'm trying to calculate possible number of combinations for a given domain name. I took stats facts here :
https://webmasters.stackexchange.com/a/16997
I have a very poor, elementary level of math so I've got this address from a friend to ask this. If someone could write me a formula how to calculate this or give me exact number or any useful information that would be great.
 A: You wrote that a word could be up to $63$ characters. I take it that means maybe $1$ character, maybe $2$, and so on up to $63$. (Mathematicians would also count the empty word, no characters, as a word, but I am assuming you do not wish to do so.)  
Let's change the rules slightly, as follows. (i) Every word is the full $63$ characters long; (ii) hyphen cannot be the last character. (Note we said nothing about first character.)
There are just as many words with this changed definition as there are with the definition you gave. For after we make a word using the new rules, we erase all the leading hyphens. 
The last character can be chosen in $36$ ways. The first $62$ can be chosen in $37^{62}$ ways, for a total of $(36)(37^{62})$. 
Remark: The answer assumes description as given, so allows multiple hyphens as in $7-b9--4x$, a total of $3$ hyphens, two of them consecutive. If we change the rules (at most one hyphen, or consecutive hyphens not allowed) the count changes. 
A: The first and last letters can only be one of $26+10=36$ characters.  The remaining 61 characters can be one of $36+1=37$ (+1, because the hyphen is an option), so the total number of possible domain names is:
$$36^{2}37^{61}$$
This is assuming that the address is exactly 63 characters long. To account for any length less than or equal to 63 is a bit trickier:
First of all, an address of length 1 can be any character except a hyphen (because the hyphen is both the first and last character. So it can be one of 36 characters (making the total amount 36). For one of length 2, it once again neither can be a hyphen, so the total number is $36^2$.
Now accounting for the rest of the numbers of characters, consider the rule of sum:
$$\sum_{k=3}^{63}36^{2}37^k$$
The grand total being:
$$36+36^2+\sum_{k=3}^{63}36^{2}37^k$$
A: For a word of length $n > 2$, there are $36^237^{n-2}$ possible combinations:
So for the total number of words up to length $n$:
$$36 + 36^2 + \sum\limits_{n=3}^{63}36^237^{n-2}$$
A: For one character names you have $36$ choices and for two characters you have $36^2$. For $n$ character names, $n \ge 3$, there are $36^2\cdot 37^{n-1}$. Summing up to $n=63$ we get $36+36^2(1+37+37^2+\ldots 37^{61}=36+36^2\frac {37^{62}-1}{37-1}=36\cdot 37^{62}$
A: Close - but 26 letters plus 10 numbers plus the hyphen is 37 characters total, so it would be
(36^2)(37^61)
Now granted, that's just the number of alphanumeric combinations; whether those combinations are actually words would require quite a bit of proofreading.
A: Hyphens really make this question a lot more difficult, because not only can a domain name not start or end with a hyphen, but it also cannot have two or more consecutive hyphens, while it can theoretically have up to thirty non-consecutive hyphens. I'm going to ignore the hyphen issue in my answer:
Letters from a to z: 26 values
Numbers from 0 to 9: 10 values
So for every letter of the domain name, it can take one of 36 values. A domain name can have a length of [1,63]. Therefore, there are 36 different domain names of length 1, $36^2$ different domain names of length 2, ... $36^{63}$ different domain names of length 63.
So we can express the number of domain names as: 
$\sum\limits_{n=1}^{63} 36^n$
