Statistics of name length

Question

Let's say a name has $26$ possibilities of characters. The maximum length of characters is $20$, the minimum is $4$. What is the statistics of a name?

I don't know how to account for the minimum of $4$ and maximum of $20$

I know if they were all maximum names it would be

$26^{20}$

Answer 1

There would be

• $26^4$ names with $4$ characters
• $26^5$ names with $5$ characters
• $26^6$ names with $6$ characters
• ...
• $26^{20}$ names with ${20}$ characters.

So, there would be $$\sum_{i=4}^{20}(26^i)=26^4+26^5+26^6+...+26^{20}=\frac{26^4(26^{17}-1)}{25}$$ possibilities for a name.