Suppose $A_1,A_2,\dots ,A_{30}$are thirty sets each having 5 elements and $B_1,B_2,\dots ,B_n$ are n sets each with 3 elements let $\underset{i=1}{\overset{30}{\cup }} A_i=\underset{j=1}{\overset{n}{\cup }} B_i=S$ and each element of $S$ belongs to exactly 10 of the $A_i^{\mathrm{\prime }}s$ and exactly 9 of the $B_i^{\mathrm{\prime }}s.$Then n is equal to 

$S=A_1\cup A_2\cup \dots \cup A_{30}=B_1\cup B_2\cup \dots \cup B_n$

Since each $A_i$ contains 5 elements, number of elements in$S$ (with repetitions) = 5 x 30 = 150

But each element of $S$ appears exactly 10 times.

$\Rightarrow$ Number of distinct elements in $S=\frac{150}{10}=15$

Similarly by other union of $B_i^{\mathrm{\prime }}{\mathrm{s}}_{\mathrm{\prime }}$ number of distinct elements in $S=\frac{3n}{9}$

Now, $\frac{3n}{9}=15\Rightarrow n=45$