In a survey of 100 MBA aspirants: 55 prepare for SNAP 50 prepare for CAT 45 prepare for XAT 25 prepare for both SNAP and CAT 20 prepare for both CAT and XAT 15 prepare for both SNAP and XAT 10 prepare for all three exams. How many students prepare for exactly one exam?

Solution

ATQ, Let S, C, X denote sets of students preparing for SNAP, CAT, XAT. Given: |S| = 55, |C| = 50, |X| = 45 |S ∩ C| = 25, |C ∩ X| = 20, |S ∩ X| = 15 |S ∩ C ∩ X| = 10 First, find those in exactly two sets: Exactly S∩C only = 25 − 10 = 15 Exactly C∩X only = 20 − 10 = 10 Exactly S∩X only = 15 − 10 = 5 Now, those only in a single set: Only S = 55 − (15 + 5 + 10) = 55 − 30 = 25 Only C = 50 − (15 + 10 + 10) = 50 − 35 = 15 Only X = 45 − (10 + 5 + 10) = 45 − 25 = 20 Total with exactly one exam = 25 + 15 + 20 = 60. (Check: Total in at least one = 60 (exactly one) + 30 (exactly two) + 10 (all three) = 100.)