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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 elements of S belongs to exactly 10 of the Ai and exactly 9 of the Bj. Then n is equal to-

A
35
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B
45
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C
55
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D
65
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Solution

The correct option is A 45
A0,A1,.............,A30 each of 5 elements
B1,B2,B3...........n each of 3 elements
The number of elements in the union of the A sets is 5(30)r where 'r' is the number repeats likewise the number of elements in the B sets 3nrB
Each element in the union (in5) is repeated 10 times in A which means if x was the real number of elements in A (not counting repeats) then q out of those 10 should be thrown away or 9x .likewise on the B side 8x of those elements should be thrown away So, 1509x=3n8x
n=503x
Now in figure out what x is we need to use the fact that the union of a group of sets contains every member of each sets . If every element in 'S' is repeated 10 times that means every element in the union of the n's is repeated 10 times .
This means that 10/1015 is the number of in the A's without repeats counted (same for the B's aswell ) So now
50153=n
n=45
Subset:- A proper subset is nothing but it contain atleast one more element of main set .
Ex:{3,4,5} is a set then the possible subsets are
{3},{4},{5},{1,5},{3,4}

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