Suppose A1,A2,…,A30 are thirty sets each having 5 elements and B1,B2,…,Bn are n sets each with 3 elements.
Let ∪i=130 Aj=∪j=1n Bj=S and each element of S belongs to exactly 10 of the Ai 's and exactly 9 of the Bj 's. Then n is =
15
3
45
35
If elements are not repeated, then number of elements in A1∪A2∪A3…∪A30 is 30×5
But each element is used 10 times, so
S=30×510=15
If elements in B1,B2,…,Bn are not repeated, then total number
of elements is 3n but each element is repeated 9 times, so
S=3n9⇒ 15=3n9⇒ n=45