Question

Suppose A₁, A₂, A₃, ..., A₃₀ are thirty sets each having 5 elements with no common elements across the sets and B₁, B₂, ..., B_n are n sets each with 3 elements with no common elements across the sets. Let \(\rm \displaystyle\bigcup^{30}_{i = 1} A_i = \displaystyle\bigcup^n_{j = 1} B_j = S\) and each elements of S belongs to exactly 10 of the A_i's and exactly 9 of the B_j's. Then n is equal to
1. 15
2. 30
3. 40
4. 45

Answer 1

Correct Answer - Option 4 : 45

Calculation:

Given: A1, A2, A3, ..., A30 are thirty sets each having 5 elements

A₁ = A₂ = .... = A₃₀ = 5

\(\rm \displaystyle\bigcup^{30}_{i = 1} A_i =A_1 + A_2 + ... +A_{30} = 30 × 5 =150\)

Given: Elements of S belongs to exactly 10 of the Ai's 

Element of S = \(\frac {150}{10} = 15\)

\(\rm \Sigma B_j\) = 9 × 15 = 135

Given: B1, B2, ..., Bn are n sets each with 3 elements 

\(\rm \displaystyle\bigcup^n_{j = 1} B_j = S\)

3B_j = 135

∴ B_j = 45