Suppose .${\mathrm{A}}_1,{\mathrm{A}}_2,\dots ,{\mathrm{A}}_{30}$ are thirty sets each having 5 elements and ${\mathrm{B}}_1,{\mathrm{B}}_2,\dots ,{\mathrm{B}}_{\mathrm{n}}$ are n sets each having 3 elements. let $\underset{\mathrm{i}=1}{\overset{30}{\cup }}{\mathrm{A}}_{\mathrm{i}}=\underset{\mathrm{j}=1}{\overset{\mathrm{n}}{\cup }} {\mathrm{B}}_{\mathrm{j}}=\mathrm{S}$ and each elements of S belongs to exactly 10 of A_i' s and exactly 9 of B_i ' s The value of n is equal to

If elements are not repeated, then number of elements in ${\mathrm{A}}_1\cup {\mathrm{A}}_2\cup {\mathrm{A}}_3\cup \dots \cup {\mathrm{A}}_{30}\text{ is }30\times 5$. But each element is
used 10 times, so
$\mathrm{S}=\frac{30\times 5}{10}=15----\left(\mathrm{i}\right)$
Similarly, if elements in ${\mathrm{B}}_1,{\mathrm{B}}_2,\dots ,{\mathrm{B}}_{\mathrm{n}}$ are not repeated, then total number of elements is 3n but each element is
repeated 9 times, so

$\begin{array}{l}\mathrm{S}=\frac{3\mathrm{n}}{9}\\ \Rightarrow 15=\frac{3\mathrm{n}}{9} \mathrm{from} \mathrm{Eq}. \left(\mathrm{i}\right)\\ \Rightarrow \mathrm{n}=45\end{array}$