Suppose that $A_1,A_2,A_3,...,A_{50}$ are fifty sets, each containing 6 elements and $B_1,B_2,B_3,...,B_n$ are $n$ sets, each containing 3 elements. If $\underset{i=1}{\overset{50}{U}}{A}_i=S=\underset{j=1}{\overset{n}{U}}{B}_i$ and each element of$S$ belongs to exactly $10$ of the $A_i\text{'}s$ and exactly $9$ of the  $B_j\text{'}s$ then $n=$

The number of elements in the set $S$ contributed by the sets $A_i\text{'}s$ is  $\frac{50\times 6}{10}=30$

The number of elements in the set $S$ contributed by the sets $B_i\text{'}s$ is  $\frac{n\times 3}{9}=\frac{n}{3}$

The above two numbers are equal, because both represents the number of elements in the same set $S$

Hence, $\frac{n}{3}=30\Rightarrow n=\boxed{\mathbf{90}}$