2007 AMC 10B Problem 8

On the trip home from the meeting where this AMC10 was constructed, the Contest Chair noted that his airport parking receipt had digits of the form bbcac, where 0 \le a < b < c \le 9, and b was the average of a and c. How many different five-digit numbers satisfy all these properties?

\textbf{(A) } 12 \qquad\textbf{(B) } 16 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 20 \qquad\textbf{(E) } 25

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