In a group of persons travelling in a bus, 6 persons can speak Tamil, 15 can speak Hindi and 6 can speak Gujarati. In that group none can speak any other language. If 2 persons in the group can speak two languages only and one person can speak all the three languages, then how many persons are there in the group ?
- (a) 21
- (b) 22
- (c) 23 ✓ UPSC's answer
- (d) 24
Why the answer is (c)
• Let the number of people speaking only Tamil, only Hindi, and only Gujarati be $T$, $H$, and $G$ respectively.
• Let the number of people speaking exactly two languages be 2, and the number speaking all three be 1.
• The total count for each language is the sum of those speaking only that language, those speaking it with one other, and those speaking all three.
• Summing the language counts: $6 + 15 + 6 = 27$.
• This sum equals $(T + H + G) + 2 \times 2 + 3 \times 1$ because people speaking two languages are counted twice and those speaking three are counted three times.
• Solving for the total number of people: $27 = (T + H + G) + 4 + 3 \Rightarrow T + H + G = 20$.
• Total persons = (Only one language) + (Exactly two languages) + (All three languages) = $20 + 2 + 1 = 23$.
Why the other options are wrong
- (a) 21
- Option (a) is incorrect because it likely results from subtracting the intersection counts incorrectly or miscalculating the sum of unique individuals.
- (b) 22
- Option (b) is incorrect because it fails to account for the correct weighting of the multi-lingual speakers in the total language count equation.
- (d) 24
- Option (d) is incorrect because it overestimates the number of people by not properly subtracting the multiple counts of the bilingual and trilingual individuals.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2015, held on 23 August 2015. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.