Question

Suppose A₁, A₂, ..., A₃₀ are thirty sets each having 5 elements and B₁, B₂, ..., Bn are n sets each with 3 elements, let \[\bigcup\limits_{i=1}^{30} A_{i} = \bigcup\limits_{j=1}^{n} B_{j}\] = 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 ______.

Options

• 15

• 3

• 45

• 35

Answer 1

Suppose A₁, A₂, ..., A₃₀ are thirty sets each having 5 elements and B₁, B₂, ..., Bn are n sets each with 3 elements, let \[\bigcup\limits_{i=1}^{30} A_{i} = \bigcup\limits_{j=1}^{n} B_{j}\] = 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 45.

Explanation:

Given: \[\bigcup\limits_{i=1}^{30} A_{i} = \bigcup\limits_{j=1}^{n} B_{j}\] 

To find: value of n

Since elements are not repeating, number of elements in A₁∪ A₂∪ A₃∪ ………∪ A₃₀ is 30 × 5

But each element is used 10 times

So, 10 × S = 30 × 5

⇒ 10 × S = 150

⇒ S = 15

Since elements are not repeating, number of elements in B₁∪ B₂∪ B₃∪ ………∪ B_n is 3 × n

But each element is used 9 times

So, 9 × S = 3 × n

⇒ 9 × S = 3n

⇒ S = `"n"/3`

⇒ `"n"/3` = 15

⇒ n = 45

Hence, the value of n is 45.