I am trying to calculate the number of possible substrings with the maximum length n in the dictionary of size $r$. I would think that it should be the number of total subsets (of all sizes) in set of $3$: $2^3=8$ but in the dictionary $\{A,B,C\}$ for example, I've counted $19$:
AAA BBB CCC ABC CBA BCA BAC AA AB AC BB BA BC CC CA CB A B C
Where am I wrong?
Suppose your alphabet has $n$ letters. Then there will be $ n^k$ words of length $k$. Sum that from $1$ to $r$ if $r$ is the maximum length of a word.
In your question you missed many of the three letter words (for example, AAB). If you write them in alphabetical order you will see the pattern.
Your guess about $2^3$ is right when you are counting subsets, but in a dictionary the order in which the letters appear in a word matters.
Related: Finding total possible permutations of n elements, in strings of n length (Math-Challenged person!)