VTRMC 2006 Problem 2

Let $S(n)$ denote the number of sequences of length $n$ formed by the three letters $A,B,C$ with the restriction that the $C$ 's (if any) all occur in a single block immediately following the first $B$ (if any). For example $ABCCAA$ , $AAABAA$ , and $ABCCCC$ are counted in, but $ACACCB$ and $CAAAAA$ are not. Derive a simple formula for $S(n)$ and use it to calculate $S(10)$ .