2007 AMC 8 Problem 24

A bag contains four pieces of paper, each labeled with one of the digits "1, 2, 3" or "4", with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. What is the probability that the three-digit number is a multiple of 3?

\textbf{(A)} \frac{1}{4} \qquad \textbf{(B)} \frac{1}{3} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{2}{3} \qquad \textbf{(E)} \frac{3}{4}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.