You can solve the general problem using linear algebra. First, construct a DFA for the language $S^*$. Then, describe its transition function in matrix form: if the state space is $Q$, construct a $Q \times Q$ matrix $A$ in which $A(s_1,s_2)$ is the number of letters that would cause a transition from $s_1$ to $s_2$. If $s$ is the starting state, let $1_s$ be the corresponding vector, and let $j$ be the all-one vector. The number of words of length exactly $n$ is $1_s A^n j$ (or perhaps $j A^n 1_s$ if I mixed things up). You can use the spectral decomposition of $A$ to get a formula, or you can calculate it quickly using repeated squaring.
That is the general theory. In your case, it's not too hard to enumerate all words of length up to $7$ by hand. I challenge you to do that.