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Question

A1,A2,...,A30are 30sets, each having 5elements and B1,B2,...,Bnare n sets each with 3elements. If i=130Ai=j=1nBj=S and each element of S belongs to exactly 10 of the Ai's and exactly 9 of the Bj's, then the value of n is


A

15

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B

3

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C

45

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D

None of these

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Solution

The correct option is C

45


Explanation for the correct option:

Finding the value of n:

Given, A's are 30 sets with five elements each.

n=130nAi=5×30

n=130nAi=1501 [given]

Let m distinct elements in S and each element of S belongs to exactly 10 of the Ai's then,

n=130nAi=10m2 [given]

From Equations (1) and (2) we get,

10m=150m=15010m=15

Similarly,

j=1nnBj=3n3 [given]

j=1nnBj=9m4 [given]

Comparing equations (3) and (4) we get

3n=9mn=9m3n=3mn=3×15m=15n=45

Hence, the value of n is 45.

Therefore, the correct answer is option (C).


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