If n(A ∩ B) = 5, n(A ∩ C) = 7 and n(A ∩ B ∩ C) = 3, then the minimum possible value of n(B ∩ C) is ____________.

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Solution

If n(A ∩ B) = 5
n(A ∩ C) = 7
n(A ∩ B ∩ C) = 3
Then the minimum possible value of n(B ∩ C)
Since n(A ⋃ B ⋃ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(C ∩ A) + n(A ∩ B ∩ C )
Since A ∩ B ∩ C ≤ B ∩ C
⇒ n(A ∩ B ∩ C) ≤ n(B ∩ C)
⇒ 3 ≤ n(B ∩ C)
∴ minimum possible value of n(B ∩ C) = 3

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