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 having 3 elements. Let each element of S belongs to exactly 10 of $A_{i}^{'}$ s and exactly 9 of $B_{i}^{'}$ s. Then, find the value of n.

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Solution

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 times.

So, $S = \frac{30 \times 5}{10} = 15 . . . (i)$

Similarly, if elements in $B_{1}, B_{2}, . . ., B_{n}$ are not repeated, then total number of elements in 3n, but each element is repeated 9 times.

So, $S = \frac{3 n}{9}$

$\Rightarrow 15 = \frac{3 n}{9}$ [from Eq. (i)]

$\Rightarrow n = \frac{15 \times 9}{3} = 45$

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