Question

Suppose sets ${A_i}\left( {i = 1,2,....,60} \right)$ each set having $12$ elements and set ${B_j}\left( {j = 1,2,3,....,n} \right)$ each set having $4$ elements let $\mathop U\limits_{i = 1}^{60} {A_i} = \mathop U\limits_{j = 1}^n {B_j} = C$ and each element of $C$ belongs to exactly $20$ of ${A_i}'s$ $18$of ${B_j}'s$ then $n$ is equal to
1. $162$
2. $36$
3. $60$
4. $120$

Answer 1

Hint: In this question, we are given that there are two sets ${A_i}\left( {i = 1,2,....,60} \right)$ and each set has $12$ elements .Also, ${B_j}\left( {j = 1,2,3,....,n} \right)$ which has $4$ elements. First step is to find the number of elements in the total sets by multiplying the number of sets and number of elements in one set. Then use the condition $\mathop U\limits_{i = 1}^{60} {A_i} = \mathop U\limits_{j = 1}^n {B_j} = C$ where you will two values and equate them.

Formula Used:
Formula to find the elements –
Number of elements in the set $ = $Number of sets $ \times $ Number of elements in one set

Complete step by step Solution:
Given that,
Number of elements in one set of ${A_i} = 12$
Total number of sets in ${A_i} = 60$
Total number of elements in $60$ sets $ = 60 \times 12 = 720$
Number of elements in one set of ${B_j} = 4$
Total number of sets in ${B_j} = n$
Total number of elements in $n$ sets $ = n \times 4 = 4n$
Also given each element of $C$ belongs to exactly $20$ of ${A_i}'s$ $18$ of ${B_j}'s$,
Therefore,
\[n\left( C \right) = \dfrac{1}{{20}}\mathop U\limits_{i = 1}^{60} {A_i}\]
\[ = \dfrac{1}{{20}}\left( {720} \right)\]
\[ = 36 - - - - - \left( 1 \right)\]
\[n\left( C \right) = \dfrac{1}{{18}}\mathop U\limits_{j = 1}^n {B_j}\]
\[ = \dfrac{1}{{18}}\left( {4n} \right) - - - - - \left( 2 \right)\]
From equation (1) and (2),
$\dfrac{{4n}}{{18}} = 36$
Cross multiplying both sides,
$4n = 648$
$n = 162$

Hence, the correct option is 1.

Note: The key concept involved in solving this problem is a good knowledge of sets. Sets are an organized collection of objects in mathematics that can be represented in set-builder or roster form. Students must know that to solve such a question where the condition is given that each element of $C$ belongs to exactly $20$ of ${A_i}'s$ $18$of ${B_j}'s$ divide the total number of elements of set by the number to which the third element belongs.