Answer
$18278$
Work Step by Step
We know that the number of arrangements of $n$ objects in $r$ slots (where in a slot only $1$ of the $n$ elements can be put) is: $n^r$.
Thus the number of company names possible with $1$ letter: $26^1=26.$
The number of company names possible with $2$ letters: $26^2=676.$
The number of company names possible with $3$ letters: $26^3=17576.$
Thus the total is:$26+676+15576=18278.$
