Suppose A1, A2, …, A30 are thirty sets each having 5 elements and B1, B2, …, Bn are n sets each with 3 elements, let
$\bigcup_{i=1}^{30} A_{i}=\bigcup_{j=1}^{n} B_{j}=S$
and each element of S belongs to exactly 10 of the Ai’s and exactly 9 of the B,’S. then n is equal to
A. 15
B. 3
C. 45
D. 35
According to the question,
$U_{i=1}^{30} A_{i}=U_{j=1}^{n} B_{j}=S$
Since elements are not repeating, number of elements in A1∪ A2∪ A3∪ ………∪ A30 = 30 × 5
Now, since each element is used 10 times
We get,
10 × S = 30 × 5
⇒ 10 × S = 150
⇒ S = 15
Since elements are not repeating, number of elements in B1∪ B2∪ B3∪ ………∪ Bn = 3 × n
Now, since each element is used 9 times
We get,
9 × S = 3 × n
⇒ 9 × S = 3n
⇒ S = n/3
⇒ n/3 = 15
⇒ n = 45
Therefore, the value of n is 45
Hence, Option (C) 45, is the correct answer.