AMC 8 2007 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 $3$? $\textbf{(A)}\ \frac{1}{4} \textbf{(B)}\ \frac{1}{3} \textbf{(C)}\ \frac{1}{2} \textbf{(D)}\ \frac{2}{3} \textbf{(E)}\ \frac{3}{4}$