Every employee of an office speaks at least one of the languages Hindi, English and Bengali. 52 employees speak Hindi, 43 speak English and 31 speak Bengali. 21 speak both Hindi and English, 14 speak both Hindi and Bengali, 11 speak both English and Bengali, and 6 speak all three languages (each ‘both’ count includes these 6). How many employees are there in the office?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 86
Adding the three language totals counts a two-language speaker twice and a three-language speaker three times. Taking away the three ‘both’ counts removes the three-language speakers three times, so they are added back once: \(n=n(H)+n(E)+n(B)-n(H\,E)-n(H\,B)-n(E\,B)+n(H\,E\,B)\).
Sum of the totals \(=52+43+31=126\)
Sum of the ‘both’ counts \(=21+14+11=46\)
Employees \(=126-46+6=86\)
Check by regions: Hindi only \(=52-21-14+6=23\), English only \(=43-21-11+6=17\), Bengali only \(=31-14-11+6=12\); two languages only: \(15,\ 8,\ 5\); all three: 6; total \(=23+17+12+15+8+5+6=86\)
