Let A = set of persons who got medals in dance. B = set of persons who got medals in dramatics. C = set of persons who got medals in music. Given, n(A) = 36 n(B) = 12 n(C) = 18 n(A U B U C) = 45 n(A ∩ B ∩ C) = 4 We know that number of elements belonging to exactly two of the three sets A, B, C = n(A ∩ B) + n(B ∩ C) + n(A ∩ C) – 3n(A ∩ B∩ C) = n(A ∩ B) + n(B ∩ C) + n(A ∩ C) – 3 × 4 ……..(i) n(A U B U C) = n(A) + n(B) + n(C) - n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C) Therefore, n(A ∩ B) + n(B ∩ C) + n(A ∩ C) = n(A) + n(B) + n(C) + n(A ∩ B ∩ C) – n(A U B U C) From (i) required number = n(A) + n(B) + n(C) + n(A ∩ B ∩ C) – n(A U B U C) – 12 = 36 + 12 + 18 + 4 – 45 – 12 = 70 – 67 = 3