Counting number of states from a regular expression

Question

Given the regular expression: $r=ab+((a+\epsilon)c^*)^*$. Let A be a non-deterministic automaton that accepts the language of r. How many states are in A? Answer the question without building A explicitly, explain how you got the answer.

I'm having trouble figuring this question out. The answer that was given is: $5*2+4*2=18$

With the explanation that for the $\epsilon$ regular expression we build an automaton with 2 states.

for each $\sigma \in \Sigma$ for the regular expression $\sigma$ we build an automaton with 2 states

For concatenating we do not add states and for each union or star operation we add 2 states.

But even with this explanation I'm not quite sure did they reach this answer.

I can understand that we have $\epsilon$, so we have 2. Plus we have $\Sigma = \{a,b,c\}$ so we do $2*3$.

Even with the stars, how do we reach $5*2$?

Also I haven't seen this kind of calculation before, are there any additional rules when trying to calculate states from a regular expression?

Answer 1

The rule for the $\sigma\in \Sigma$ must be applied each time a letter appears in the regular expression, not only once per letter : you must build a different automata each time the letter appears. If you see it as some kind of digital circuit, you do not want to use the circuit that recognizes $a$ in $ab$ for the circuit that recognizes $a$ in $a + \epsilon$.

So you have five letters that appear in $r$, that's $5*2$ states. Then you have 2 concatenations, two unions and two stars. That's $2 * 0 + 2*2+2*2 = 4*2$ states.