Question

Suppose \[A_1 , A_2 , . . . , A_{30}\] are thirty sets each having 5 elements and \[B_1 , B_2 , . . . , B_n\] are n sets each with 3 elements. Let \[\cup^{30}_{i = 1} A_i = \cup^n_{j = 1} B_j = S\] and each element of S belong to exactly 10 of the \[A_i 's\]and exactly 9 of the\[B_j 's\] then n is equal to 

पर्याय

• (a) 15      

•   (b) 3              

•  (c) 45     

•  (d) 35  

Answer 1

It is given that each set A_{i \[\left( 1 \leq i \leq contain 5 elements and} \[\cup^{30}_{i = 1} A_i = S\] 

\[\therefore n\left( S \right) = 30 \times 5 = 150\] 

But, it is given that each element of S belong to exactly 10 of the A_i's. \[\frac{150}{10} = 15\]     .....(1) 

It is also given that each set B_j 

_{\[\left( 1 \leq j \leq n \right)\] contains 3 elements and} \[\cup^n_{j = 1} B_j = S\]

\[\therefore n\left( S \right) = n \times 3 = 3n\] 

Also, each element of S belong to eactly 9 of B_j's.

∴ Number of distinct elements in S = \[\frac{3n}{9}\] 

From (1) and (2), we have 

\[\frac{3n}{9} = 15\]
\[ \Rightarrow n = 45\] 

Thus, the value of n is 45.

Hence, the correct answer is option (c).