1959 AHSME Problem 50

A club with x members is organized into four committees in accordance with these two rules:

\begin{array}{l} \text{(1)}\ \textup{Each member belongs to two and only two committees}\qquad \\ \text{(2)}\ \textup{Each pair of committees has one and only one member in common} \end{array}

Then x:

\begin{array}{l} \textbf{(A)} \ \textup{cannot be determined} \qquad \\ \textbf{(B)} \ \textup{has a single value between 8 and 16} \qquad \\ \textbf{(C)} \ \textup{has two values between 8 and 16} \qquad \\ \textbf{(D)} \ \textup{has a single value between 4 and 8} \qquad \\ \textbf{(E)} \ \textup{has two values between 4 and 8} \end{array}

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