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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=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 45
Let 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
A1A2A3.....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
B1B2B3.....Bn=3n.
But each element in Bis is repeated 9 times.
m=3n93n9=15n=45

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