Question 8: In a group of 500 employees, 250 can speak Urdu, 150 can speak English, 50 can speak punjabi, 40 can speak Urdu and English, 30 can speak both English and punjabi, and 10 can speak Urdu and Punjabi. How many can speak all three languages?
Solution:
Let, \(U=\text{Urdu speakers}\; ;\)\(\;E=\text{English speakers}\; ;\)\(\;P=\text{Punjabi speakers}\)
\(n(U \cup E \cup P)=500\)
\(n(U)=250\)
\(n(E)=150\)
\(n(P)=50\)
\(n(U\cap E)=40\)
\(n(E\cap P)=30\)
\(n(U \cap P)=10\)
\(n(U\cap E \cap P)=?\)
We know that
\(n(U\cup E\cup P)=\ \)\(n(U)+n(E)+n(P)\ \)\(−\ n(U\cap E)−n(E\cap P)\ \)\(−\ \)\(n(U\cap P)+n(U\cap E\cap P)\)
\(500=\ \)\(250+150+50\ \)\(−\ 40−30\ \)\(−\ \)\(10+n(U\cap E\cap P)\)
\(500=\ \)\(370+n(U\cap E\cap P)\)
\(n(U\cap E\cap P)=500-370\)
\(n(U\cap E\cap P)=130\)
\(\boxed{130 \text{ employees can speak all three languages}}\)