1997 Pascal Problem 19

Three balls numbered 1, 2, and 3 are placed in a bag. A ball is drawn from the bag and the number is recorded. The ball is returned to the bag. After this has been done three times, what is the probability that the sum of the three recorded numbers is less than 8?

\textbf{(A)}\ \frac{23}{27}\quad \textbf{(B)}\ \frac{5}{7}\quad \textbf{(C)}\ \frac{5}{9}\quad \textbf{(D)}\ \frac{8}{9}\quad \textbf{(E)}\ \frac{22}{27}

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Pascal contests here.