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(C) Let $|S|$ be the number of elements in the set $S$.Given that each of the $30$ sets $A_i$ has $5$ elements,the sum of the number of elements in al

Vedclass · Accepted Answer

$45$

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(C) Let $|S|$ be the number of elements in the set $S$.Given that each of the $30$ sets $A_i$ has $5$ elements,the sum of the number of elements in all $A_i$ is $30 	imes 5 = 150$.Since each element of $S$ belongs to exactly $10$ of the $A_i$'s,we have $|S| 	imes 10 = 150$,which implies $|S| = 15$.Similarly,for the sets $B_j$,each of the $n$ sets has $3$ elements,so the sum of the number of elements in all $B_j$ is $n 	imes 3 = 3n$.Since each element of $S$ belongs to exactly $9$ of the $B_j$'s,we have $|S| 	imes 9 = 3n$.Substituting $|S| = 15$,we get $15 	imes 9 = 3n$.$135 = 3n \Rightarrow n = 45$.

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