# DFA for language of all strings avoiding 'aa'

I'm trying to draw a dfa for this description

The set of strings over {a, b, c} that do not contain the substring aa,

current issue i'm facing is how many states to start with, any help how to approach this problem?

For an alphabet of $$\{a, b, c\}$$ constructing a DFA that accepts all strings not containing the substring 'aa' tells you several things about the number of states you need. Firstly, and this is true for any DFA, you need at least one accepting state. Since your DFA is meant to filter out some strings, it requires a 'trap' or 'dead' state, a state that may never reach an accepting state once reached. Now clearly the strings 'a', 'b', 'c', and '' (empty string) do not contain the substring 'aa', therefore we know our initial state must also be an accepting state, which eliminates the next obvious choice for a new state.