Step 1: Understanding the Question:
The topic of this problem is Set Theory, specifically focusing on the Principle of Inclusion-Exclusion for three finite sets. The objective is to find the cardinality of the union of three groups (Odia, English, and Hindi speakers). The term "at least one language" mathematically translates to finding the total number of unique elements present across all three sets combined, ensuring that individuals who speak multiple languages are not counted more than once. There is a slight typo in the question text where "Odia and English" is repeated; logically, the second instance refers to the third pair of languages, which we will treat as Odia and Hindi to match the standard calculation for 131.
Step 2: Key Formulas and approach:
To solve problems involving the union of three sets, we use the Inclusion-Exclusion Principle formula:
\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - [n(A \cap B) + n(B \cap C) + n(C \cap A)] + n(A \cap B \cap C) \]
Our approach involves identifying each individual set's value, the values of their pairwise intersections, and finally the intersection of all three sets. By substituting these into the formula, we systematically add the totals and subtract the overlaps to arrive at the final count.
Step 3: Detailed Explanation:
Let the set of Odia speakers be $O$, English speakers be $E$, and Hindi speakers be $H$.
From the provided data, we identify the individual set sizes: $n(O) = 86$, $n(E) = 64$, and $n(H) = 42$.
We identify the dual-language speakers (intersections): $n(O \cap E) = 39$, $n(E \cap H) = 21$, and $n(O \cap H) = 17$.
We identify the speakers of all three languages: $n(O \cap E \cap H) = 16$.
Substitute these values into the Inclusion-Exclusion formula: $86 + 64 + 42 - (39 + 21 + 17) + 16$.
First, calculate the sum of individuals in each set: $86 + 64 + 42 = 192$.
Next, calculate the sum of the pairwise intersections: $39 + 21 + 17 = 77$.
Subtract the intersections from the individual sums: $192 - 77 = 115$.
Finally, add back the intersection of all three sets to account for the triple-counting correction: $115 + 16 = 131$.
Step 4: Final Answer:
The number of persons who know at least one of the three languages in the group is 131.