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:

Always draw a Venn diagram and start filling values from the innermost intersection (all three sets) outwards to avoid double counting.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem is based on the Principle of Inclusion-Exclusion for three sets.
Let the sets of people who know Odia, English, and Hindi be \(O\), \(E\), and \(H\) respectively.
Based on standard problem structures covering all pairwise intersections, the second instance should logically be "17 know Odia and Hindi".
We will proceed with this corrected assumption.


Step 2: Key Formula or Approach:
The formula for the union of three sets is:
\[ |O \cup E \cup H| = |O| + |E| + |H| - |O \cap E| - |E \cap H| - |O \cap H| + |O \cap E \cap H| \]

Step 3: Detailed Explanation:
From the given data, we have the following set sizes:
\( |O| = 86 \)
\( |E| = 64 \)
\( |H| = 42 \)
\( |O \cap E| = 39 \)
\( |E \cap H| = 21 \)
\( |O \cap H| = 17 \) (assuming the logical typo correction).
\( |O \cap E \cap H| = 16 \)
Substitute these values directly into the inclusion-exclusion formula:
\[ |O \cup E \cup H| = 86 + 64 + 42 - 39 - 21 - 17 + 16 \] \[ |O \cup E \cup H| = 192 - 77 + 16 \] \[ |O \cup E \cup H| = 115 + 16 = 131 \]

Step 4: Final Answer:
The total number of persons who know at least one language is 131.

Answer 2

Instead of plugging the given numbers straight into the inclusion-exclusion formula, we can build up the Venn diagram region by region, which also shows exactly how many people fall into each combination of languages.

Let \(O\), \(E\), and \(H\) be the sets of people who know Odia, English, and Hindi. We are given \(|O|=86\), \(|E|=64\), \(|H|=42\), \(|O\cap E|=39\), \(|E\cap H|=21\), \(|O\cap H|=17\), and \(|O\cap E\cap H|=16\).

First, find the region for each pair of languages excluding the people who also know the third:

\[ \text{Only } O \text{ and } E = 39-16 = 23 \] \[ \text{Only } E \text{ and } H = 21-16 = 5 \] \[ \text{Only } O \text{ and } H = 17-16 = 1 \]

Next, find the region for each language alone (excluding anyone who also knows another language):

\[ \text{Only } O = 86-(23+1+16) = 46 \] \[ \text{Only } E = 64-(23+5+16) = 20 \] \[ \text{Only } H = 42-(5+1+16) = 20 \]

Now the Venn diagram is fully filled in with seven non-overlapping regions: only O (46), only E (20), only H (20), O and E only (23), E and H only (5), O and H only (1), and all three (16).

Adding every one of these non-overlapping regions gives the total number of people who know at least one language:

\[ 46+20+20+23+5+1+16 = 131 \]

Therefore, the correct answer is 131.