Question

# Suppose A1,A2,....,A30 are thirty sets, each having 5 elements and B1,B2,.....,Bn are n sets, each with 3 elements. Let ⋃30i=1Ai=⋃ni=1Bj=S. Each element of S belongs to exactly 10 of the Ais and exactly 9 of the Bis. Then n is equal to

A
15
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B
3
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C
45
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D
50
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Solution

## The correct option is C 45Let m be the number of elements in S. Given that each of the Ais have 5 elements. If no element in Ais is repeated, then the number of elements in A1∪A2∪A3.....∪A30=(30×5). But each element of the Ais is used 10 times. ∴ m=30×510=15. If the elements in B1,B2,.....,Bn, are not repeated, then the total number of elements in B1∪B2∪B3.....∪Bn=3n. But each element in Bis is repeated 9 times. ∴ m=3n9⇒3n9=15⇒n=45

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