Question

$A_1,A_2,\cdots ,A_{30}$ are $30$ sets, each having $5$ elements and $B_1,B_2,\cdots ,B_n$ are $n$ sets each with $3$ elements. If $\bigcup _{i=1}^{30}{A}_i=\bigcup _{j=1}^n{B}_j=S$ and each element of $S$ belongs to exactly $10$ of the $A_i$ 's and exactly $9$ of the $B_j$ 's, then the value of $n$ is

• A. $115$
• B. $35$
• C. $45$
• D. $25$

Answer 1

The correct option is C $45$

Given, $A$'s are $30$ sets with five elements each.
$\sum _{i=1}^{30}n\left(A_i\right)=5\times 30=150\cdots \left(1\right)$

Let $m$ distinct elements in $S$ and each elements of $S$ belongs to exactly $10$ of the $A_i$ 's, then $\sum _{i=1}^{30}n\left(A_i\right)=10m\cdots \left(2\right)$
From Eqs. $\left(1\right)$ and $\left(2\right),m=15$

Similarly, $\sum _{j=1}^{n}n\left(B_j\right)=3n$
$\sum _{j=1}^{n}n\left(B_j\right)=9m$
$\therefore 3n=9m$
$\Rightarrow n=\frac{9m}{3}=3\times 15=45$