Question

Suppose $A_1,A_2.....,A_{30}$ are thirty sets each with five elements and $B_1,B_2.....,B_n$ are 'n' sets each with three elements.
Let $\underset{i=1}{\stackrel{30}{\cup }}{A}_i=\underset{j=1}{\stackrel{n}{\cup }}{B}_j=S.<br>$ Assume that each element of S belongs to exactly 10 of ${A_I}^{\prime }s$ and exactly 9 of ${B_j}^{\prime }s,$ then the value of $n$ is

• A. $90$
• B. $15$
• C. $9$
• D. 45

Answer 1

Answer: (D)

If elements are not repeated then number of elements in
$A_1\cup A_2\cup A_3\cup ......\cup A_{30}$
is $30\times 5.$
but each element is used 10 time.
$\therefore$ $S=\frac{30\times 5}{10}=15$ ...(i)
Similarly, if elements in $B_1,B_2.....B_n$
are not repeated, then total number of elements is 3n but each elements is repeated 9 times.
$S=\frac{3n}{9}$
$\Rightarrow$ $15=\frac{3n}{9};$
[from Eq. (i)]
$\therefore$ $n=45$