$\text{ Suppose }{A}_1,A_2,\dots ,A_{30}\text{ are thirty sets each having }5\text{ elements}$$\mathrm{and} B_1,B_2,\dots ,B_n\text{ are }n\text{ sets each with }3\text{ elements. }$

$\text{ Let }\underset{i=1}{\overset{30}{\cup }} A_i=\underset{i=1}{\overset{n}{\cup }} B_j=S\text{ and each element of }S\text{ belongs to exactly }10\text{ of the }{A}_i\text{ 's and exactly }9\text{ of the }{B}_j\text{ 's. }$

$\mathrm{Then} n\text{ is equal to}$

$\text{ If elements are not repeated, then number of elements in }$

$A_1\cup A_2\cup A_3,\dots \cup A_{30}\text{ is }30\times 5$

$\text{ But each element is used }10\text{ times, so }$

$S=\frac{30\times 5}{10}=15$

$\text{ If elements in }{B}_1,B_2,\dots ,B_n\text{ , are not repeated, then total number }$

$\text{ of elements is }3n\text{ . But each element is repeated }9\text{ times, so }$

$\begin{array}{l}S=\frac{3n}{9}\\ \Rightarrow 15=\frac{3n}{9}\\ \Rightarrow n=45\end{array}$