Suppose A1,A2,…,A30are thirty sets each having 5 elements and B1,B2,…,Bn are n sets each with 3 elements let ∪i=130 Ai=∪j=1n Bi=S and each element of S belongs to exactly 10 of the Ai′s and exactly 9 of the Bi′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 $B_{i}^{'} s_{'}$ number of distinct elements in $S = \frac{3 n}{9}$

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

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