Question

Suppose $A_1,A_2,...,A_{3}0$ are thirty sets each having 5 elements and $B_1,B_2,...,B_n$ are n sets each with 3 elements, let $U_i^{30}=1A_i=U_j^n=1B_j=S$ and each element of S belongs to exactly 10 of the $A_i{}^{\prime }s$ and exactly 9 of the $B_i{}^{\prime }s$ then n is equal to

• A. 35
• B. 15
• C. 45
• D. 3

Answer 1

The correct option is C 45

Given : $A_i$ have 5 elements in the set and $B_i$ have 3 elements in the set

$A_1\cup A_2\cup A_3\cup ...\cup A_{3}0=S$

and

$B_1\cup B_2\cup B_3\cup ...\cup B_n=S$

If elements are not repeated, then number of elements in

$A_1\cup A_2\cup A_3\cup ...\cup A_{3}0is30\times 5$

But each element is used 10 times

So, $S=\frac{30\times 5}{10}=...\left(1\right)$

If elements in $B_1,B_2,B_3,...,B_n$ are not repeated, then total number of elements in

$B_1\cup B_2\cup B_3\cup ...\cup B_{n}is3n$

But each element is repeated 9 times

So, $S=\frac{3n}{9}$

$\Rightarrow S=\frac{n}{3}...\left(2\right)$

Equating equation (1) with (2), we get $\frac{n}{3}$$=15$

$\Rightarrow n=45$

Hence, option (C) is correct.