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Suppose $A_1, A_2, ..., A_{30}$ are thirty sets each having $5$ elements and $B_1, B_2, ..., B_n$​​​​​​​ are n sets each with $3$ elements. Let $\bigcup\limits^{30}_\text{i = 1}\text{A}_\text{i}=\bigcup\limits^{\text{n}}_\text{j = 1}\text{B}_\text{j}=\text{S}$ and each element of S belong to exactly $10$ of the $A_i^{'s}$​​​​​​​ and exactly $9$ of the $B_j^{'s}​​​​​​​$​​​​​​​, then n is equal to:

• A
  $15$
• B
  $3$
• ✓
  $45$
• D
  $35.$