Question

# Suppose A1,A2,...A30 are thirty sets each having 5 elements and B1,B2,...Bn are n sets each with 3 elements. Let 30⋃i=1Ai=n⋃i=1Bi=S and each element of S belongs to exactly 10 of the Ai and exactly 9 of the Bi. then

A
n is an odd number
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B
n is an odd prime number
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C
n has two prime factors
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D
n has only one prime factor
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Solution

## The correct options are A n is an odd number C n has two prime factorsNumber of elements in A1∪A2∪−−−−∪A30 is 30×5=150 But each element is used 10 times. So S=30×510=15 Total number of elemetns in B1∪B2∪−−−−∪ Bn=3n But each element is repeated 9 times. So, S=3n9=15 ⇒n=45

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