Question

In a group of people it was observed that 86 persons know Odia, 64 know English, 42 know Hindi, 39 know Odia and English, 21 know English and Hindi, 17 know Odia and English, and 16 persons know all the three languages. How many persons in the group know at least one language?

• A. 131
• B. 99
• C. 192
• D. None of the above

Hint:

Use the inclusion-exclusion principle to calculate the total number of elements in overlapping sets.

Answer 1

The Correct Option is A

Step 1: Add the number of people who know each individual language: \( 86 + 64 + 42 = 192 \).
Step 2: Subtract every pairwise overlap once, since people who know two languages have been counted twice in step 1: \( 192 - 39 - 21 - 17 = 115 \).
Step 3: Add back the number of people who know all three languages, since they get removed one time too many in step 2: \( 115 + 16 = 131 \).
\[ \boxed{131} \]

Answer 2

A different way to verify this is to break the group into the seven distinct, non-overlapping regions of a three-set Venn diagram, and add those regions up directly instead of using the inclusion-exclusion formula as a single expression.

1. All three languages: Given directly as 16 people.
2. Odia and English only (not Hindi): The Odia-English overlap is 39, and 16 of those also know Hindi, so only-Odia-and-English is \( 39 - 16 = 23 \).
3. English and Hindi only (not Odia): The overlap is 21, minus the 16 who know all three, giving \( 21 - 16 = 5 \).
4. Odia and Hindi only (not English): The remaining pairwise figure of 17 represents the Odia-Hindi overlap, so only-Odia-and-Hindi is \( 17 - 16 = 1 \).
5. Odia only: Total Odia speakers is 86. Removing the only-Odia-English region (23), only-Odia-Hindi region (1), and all-three region (16) leaves \( 86 - 23 - 1 - 16 = 46 \).
6. English only: Total English speakers is 64. Removing the only-Odia-English region (23), only-English-Hindi region (5), and all-three region (16) leaves \( 64 - 23 - 5 - 16 = 20 \).
7. Hindi only: Total Hindi speakers is 42. Removing the only-English-Hindi region (5), only-Odia-Hindi region (1), and all-three region (16) leaves \( 42 - 5 - 1 - 16 = 20 \).

Adding every one of these seven non-overlapping regions together: \( 46 + 20 + 20 + 23 + 5 + 1 + 16 = 131 \), matching the direct inclusion-exclusion result exactly.

Therefore, the correct answer is 131.