**Question:**

**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 toA. 15B. 3C. 45D. 35**

**Solution:**

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.