Set Theory

Suppose $P_1,P_2,\dots,P_{30}$ are thirty sets each having $5$ elements and $Q_1,Q_2,\dots,Q_n$ are $n$ sets with $3$ elements each. Let $\bigcup_{i=1}^{30}P_i=\bigcup_{j=1}^{n}Q_j=S$ and each element of $S$ belongs to exactly $10$ of the $P$’s and exactly $9$ of the $Q$’s. Then $n$ equals