Suppose
(a) 15
(b) 3
(c) 45
(d) 35
It is given that each set $A_{i}(1 \leq i \leq 30)$ contains 5 elements and $\bigcup_{i=1}^{30} A_{i}=S$.
$\therefore n(S)=30 \times 5=150$
But, it is given that each element of S belong to exactly 10 of the Ai's.
$\therefore$ Number of distinct elements in $S=\frac{150}{10}=15$ ....(1)
It is also given that each set $B_{j}(1 \leq j \leq n)$ contains 3 elements and $\bigcup_{j=1}^{n} B_{j}=S$.
$\therefore n(S)=n \times 3=3 n$
Also, each element of S belong to eactly 9 of Bj's.
$\therefore$ Number of distinct elements in $S=\frac{3 n}{9}$ ....(2)
From (1) and (2), we have
$\frac{3 n}{9}=15$
$\Rightarrow n=45$
Thus, the value of n is 45.
Hence, the correct answer is option (c).
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