AHSME 1967 Problem 38

Given a set $S$ consisting of two undefined elements "pib" and "maa", and the four postulates: $P_1$: Every pib is a collection of maas, $P_2$: Any two distinct pibs have one and only one maa in common, $P_3$: Every maa belongs to two and only two pibs, $P_4$: There are exactly four pibs. Consider the three theorems: $T_1$: There are exactly six maas, $T_2$: There are exactly three maas in each pib, $T_3$: For each maa there is exactly one other maa not in the same pid with it. The theorems which are deducible from the postulates are: $\textbf{(A)}\ T_3 \text{only} \textbf{(B)}\ T_2 \text{and} T_3 \text{only} \textbf{(C)}\ T_1 \text{and} T_2 \text{only}\\ \textbf{(D)}\ T_1 \text{and} T_3 \text{only} \textbf{(E)}\ \text{all}$