Question

Suppose A₁, A₂, …, A₃₀ are thirty sets each having 5 elements and B₁, B₂, …, B_n are n sets each with 3 elements, let 

and each element of S belongs to exactly 10 of the A_i’s and exactly 9 of the B,’S. then n is equal to

A. 15
B. 3
C. 45
D. 35

Answer 1

According to the question,

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∪ ………∪ B_n = 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.