AMC 12 2005a Problem 24

Let $P(x) = (x - 1)(x - 2)(x - 3)$. For how many polynomials $Q(x)$ does there exist a polynomial $R(x)$ of degree $3$ such that $P(Q(x)) = P(x) \cdot R(x)$?

$\textbf{(A) } 19 \textbf{(B) } 22 \textbf{(C) } 24 \textbf{(D) } 27 \textbf{(E) } 32$