solving for k yields k=3.
I will give a direct proof here and then apply it to this problem.
Since
we have
and
we get
Thus
Now lets apply the formula to this problem. N=52 and n=13. Thus,
is the initial state.
and
are final states.
where is the initial state
and
is the final state.
where is the initial state
and
is the final state.
This is a non-deterministic finite state automaton.