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 $\underset{i=1}{\mathop{\overset{30}{\mathop{\cup }}\,}}\,\,\,{{A}_{i}}=\underset{j=1}{\mathop{\overset{n}{\mathop{\cup }}\,}}\,\,\,\,{{B}_{j}}=S.$ Assume that each element of S belongs to exactly 10 of ${{A}_{I}}'s$ and exactly 9 of ${{B}_{j}}'s,$ then the value of $n$ is

• A. $90$
• B. $15$
• C. $9$
• D. 45

Answer 1

Answer: (D)

45

Answer 2

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 E (i)]
$\therefore$
$n=45$