Choose the correct answers from the given four options: Suppose $A_{1}$, $A_{2}$, ..., $A_{30}$ are thirty…
Choose the correct answers from the given four options:
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 $\cup_{i=1}^{30} A_{i} = \cup_{j=1}^{n} B_{j} = S$ and each element of $S$ belongs to exactly 10 of the $A_{i}$’s and exactly 9 of the $B$'s. Then $n$ is equal to.
15
3
45
35
Solution
Solution:
Number of elements in $A_1 \cup A_2 \cup A_3 \ldots \cup A_{30} = 30 \times 5 = 150$ (When repetition is not allowed)
But each element is repeated 10 times
$\therefore n(S) = \frac{30 \times 5}{10}$ = $\frac{150}{10}$ = 15 (i)
Number of elements in $B_1 \cup B_2 \cup B_3 \ldots \cup B_n = 3n$ (when repetition is not allowed)
But each element is repeated 9 times
$\therefore n(S) = \frac{3n}{9} = \frac{n}{3}$ (ii)
From (i) and (ii) we get
$\frac{n}{3} = 15 \Rightarrow n = 15 \times 3 = 45$
Hence, the correct option is (c).