AMC 12 2000 Problem 23

Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $1$ through $46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property- the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket?

$\textbf {(A)}\ 1/5 \textbf {(B)}\ 1/4 \textbf {(C)}\ 1/3 \textbf {(D)}\ 1/2 \textbf {(E)}\ 1$