UPSC NDA Math Model Paper 5

Let, A = Number of students awarded with tablet, B = Number of students awarded with laptop and C = Number of students awarded with smartphone.
Given: n (A ∪ B ∪ C) = 70, n (A) = 38, n (B) = 15, n (C) = 20, n (A ∩ B) = 5, n (B ∩ C) = 3 and n (A ∩ C) = 2.
As we know that, if A, B and C are 3 finite sets, then
n (A ∪ B ∪ C) = n (A) + n (B) + n (C) – n(A ∩ B) – n (B ∩ C) – n (A ∩ C) + n (A ∩ B ∩ C)
⇒ 70 = 38 + 15 + 20 – 5 – 3 – 2 + n (A ∩ B ∩ C)
⇒ 70 = 63 + n (A ∩ B ∩ C)
⇒ n (A ∩ B ∩ C) = 7
Hence, there were 7 students who were awarded with all three.