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)

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 $