Find the number of elements in the union of 4 sets A, B, C and D having 150, 180, 210 and 240 elements respectively, given that each pair of sets has 15 elements in common. Each triple of sets has 3 elements in common and A ∩ B ∩ C ∩ D = ϕ
\[ A \cup B \cup C \cup D = A + B + C + D - (A \cap B) - (A \cap C) - (A \cap D) - (B \cap C) - (B \cap D) - (C \cap D) \] \[ + (A \cap B \cap C) + (A \cap B \cap D) + (A \cap C \cap D) + (B \cap C \cap D) - (A \cap B \cap C \cap D) \] Substituting the given values: \[ = 150 + 180 + 210 + 240 - 15 - 15 - 15 - 15 - 15 - 15 + 3 + 3 + 3 + 3 + 3 - 0 \] Simplifying: \[ = 702 \]
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