Suppose A 1 ,   A 2 ,   . . . ,   A 30 are thirty sets each having 5 elements and B 1 ,   B 2 ,   . . . . ,   B n are n sets each with 3 elements. Let ∪ i = 1 30 A i = ∪ j = 1 n B j = S and each element of S belongs to exactly 10 of the A i ' s and exactly 9 of the B j ' s. Then n is equal to

Step-by-step solution

∑ i = 1 30 n A i = 5 × 30 = 150 Let S consist of m distinct elements. Since each elements of S belongs to exactly 10 of the A i ' , so, we have ∑ i = 1 30 n A i = 10 m = 150 ⇒ m = 15 Now each B i has 3 elements and each elements of S belongs to exactly 9 of the B i 's. So, we have ∑ j = 1 n n B j = 3 n = 9 m ⇒ 3 n = 9 × 15 ⇒ n = 45