$\mathbf{\underline{Answer:}\Rightarrow}$ No option matches (Answer $\mathbf{=13}$)
$\mathbf{\underline{Explanation:}\Rightarrow}$
Though everyone has already given the correct answer but let me try an easier explanation.
You can do it by hit and trial method.
The smallest string you can think of in the first attempt is
Let's say $\mathbf{x_1}$, then it should contain either $\mathbf 0 \; \text{or}\; \mathbf 1$.
So, for $\mathbf{x_1}\; \textbf{Answer} = \mathbf 2$
For $\mathbf{x_2}$, it may contain, $11, 10 \; \text{or}\; 01 \rightarrow = 3$
Similarly, for $\mathbf{x_3}$, it may contain, $111, 110, 101, 011, 010 \rightarrow = 5$
Now, you can see that it is forming a series like $2, 3, 5, ...$, and so on.
Recall, this popular series. It's nothing but the Fibonacci series.
So, here's your answer.
since, $\mathbf{f(n) = f(n-1) + f(n-2)}$, (By Fibonacci Series)
So, $\mathbf{f(5) = f(4) + f(3), ...............(1)}$
$\mathbf{f(4) = f(3) + f(2) = 5 + 3...........(2)}$
Putting this value of equation $\mathbf{(2)}$ in equation $\mathbf{(1)}$, we get:
$\mathbf{f(5) = 8 + 5}$
$\therefore \; \textbf{Answer} = 13$