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Suppose A1, A2,..., A30 are thirty sets each having 5 elements and B1, B2,...Bn,..., are n sets each having 3 elements. Let

= S   

and each elements of S belongs to exactly 10 of Ai ’s and exactly 9 of Bj ’s. The value of n is equal to 

(a) 15                  (b) 3 

(c) 45                  (d) None of these

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Correct option is (c)


If elements are not repeated, then number of elements in A1 ∪ A2 ∪... ∪ An ... is 30 x 5. But each element is used 10 times so, S = (30 × 5)/10 =15. Similarly, if elements in B1,B2,...,Bn are not repeated, then total number of elements is 3n but each element is repeated 9 times so, 

S = 3n/9 = 15 ⇒ n = 45.

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