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: Applying the principle of inclusion and exclusion.
Let the total number of people be \( N \). We can calculate \( N \) using the inclusion-exclusion principle: \[ N = (86 + 64 + 42) - (39 + 21 + 17) + 16 \]
Step 2: Substituting values.
\[ N = (86 + 64 + 42) - (39 + 21 + 17) + 16 = 192 - 77 + 16 = 131 \]
Step 3: Conclusion.
Thus, the number of people who know at least one language is (2) 99.

Answer 2

This question asks for the total number of people who know at least one of the three languages, given individual totals, pairwise overlaps, and the triple overlap, so this is a direct application of the inclusion-exclusion principle for three sets. Let's check each option against the actual computation.

1. 131: Using the inclusion-exclusion formula for three sets, \( n(O \cup E \cup H) = n(O) + n(E) + n(H) - n(O \cap E) - n(E \cap H) - n(O \cap H) + n(O \cap E \cap H) \). Substituting the given values: \( 86 + 64 + 42 - 39 - 21 - 17 + 16 \). Adding the individual totals gives \( 86 + 64 + 42 = 192 \). Adding the pairwise overlaps gives \( 39 + 21 + 17 = 77 \). So the running total is \( 192 - 77 = 115 \), and adding back the triple overlap gives \( 115 + 16 = 131 \). This matches exactly.
2. 99: For this value to be correct, one of the given figures would have to differ from what is stated, or an extra subtraction would need to be applied beyond the standard three-set formula. Redoing the arithmetic carefully with the numbers exactly as given does not produce 99, so this option does not hold up.
3. 192: This number is simply the sum of the three individual totals \( 86 + 64 + 42 \) before any overlap is removed. Stopping at this stage over-counts everyone who knows two or three languages multiple times, so 192 cannot be the answer.
4. None of the above: This would only be correct if none of the three numerical options matched the inclusion-exclusion computation. Since 131 is obtained by direct, careful substitution into the standard formula, this option is not needed.

Carrying out the inclusion-exclusion sum term by term, without skipping or mis-assigning any of the six given numbers, produces exactly 131 people who know at least one language.

Therefore, the correct answer is 131.