In a room containing 18 people, there are 15 people who speak English, 13 people who speak Hindi and 11 people who speak Kannada. 9 persons speak both English and Hindi, 9 persons speak both Hindi and Kannada whereas 9 persons speak both Kannada and English. How many speak all three languages? A. 9 B. 8 C. 7 D. 6

GATE 1998 · Discrete Mathematics · Set Theory · easy

Answer: D. 6

1. Write the three-set inclusion-exclusion formula: 18 = 15 + 13 + 11 - 9 - 9 - 9 + |E ∩ H ∩ K|
2. Solve for the triple intersection: 18 = 15+13+11 - 27 + x = 39 - 27 + x = 12 + x => x = 18 - 12 = 6.