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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=nj=1Bj=S and each element of S belongs to exactly 10 of the Ais and exactly 9 of the Bjs. Then n is equal to

A
15
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B
135
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C
45
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D
90
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Solution

The correct option is B 45
Since each A Ai has 5 elements, we have 30i=1n(Ai)=5×30=150(1)
Let S consist of m distinct elements. Since each element of S belongs to exactly 10 of the Ais, we also have 30i=1n(Ai)=10m(2)
Hence from (1) and (2), 10m=150 or m=15.
Again since each Bi has 3 elements and each element of S belongs to exactly 9 of the BiS, we have nj=1n(Bi)=3n and nj=1n(Bj)=9m
It follows that 3n=9m=9×15 [m=15]
This gives n=45

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