Question

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 ${\bigcup }_{i=1}^{30}{A}_i={\bigcup }_{i=1}^n{B}_j=S$. Each element of S belongs to exactly 10 of the $A_{i}s$ and exactly 9 of the $B_{i}s$. Then n is equal to

• A. 15
• B. 3
• C. 45
• D. 50

Answer 1

The correct option is C

45

Let $m$ be the number of elements in S. Given that each of the $A_{i}s$ have 5 elements.
If no element in $A_{i}s$ is repeated, then the number of elements in
$A_1\cup A_2\cup A_3.....\cup A_{30}=(30\times 5)$.
But each element of the $A_{i}s$ is used 10 times.
$\therefore m=\frac{30\times 5}{10}=15$.
If the elements in $B_1,B_2,.....,B_n,$ are not repeated, then the total number of elements in
$B_1\cup B_2\cup B_3.....\cup B_n=3n$.
But each element in $B_{i}s$ is repeated 9 times.
$\therefore m=\frac{3n}{9}\Rightarrow \frac{3n}{9}=15\Rightarrow n=45$