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Question

Let 50i=1Xi=ni=1Yi=T, where each Xi contains 10 elements and each Yi contains 5 elements. If each element of the set T is an element of exactly 20 of sets Xi's and exactly 6 of sets Yi's, then n is equal to

A
15
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B
30
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C
50
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D
45
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Solution

The correct option is B 30
Given: T=50i=1Xi=ni=1Yi
It is given that each set of X contains 10 elements.
n(T)=50×10=500
But it is given that each element of the set T is an element of exactly 20 of sets Xi's.
Number of distinct elements in T=50020 (1)

It is also given that each set of Y contains 5 elements.
n(T)=n×5=5n
But it is given that each element of the set T is an element of exactly 6 of sets Yi's.
Number of distinct elements in T=5n6 (2)

From (1) and (2), we get
502=5n6
n=30

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