Understanding the College Language Knowledge Problem

The problem provides information about the language knowledge of 3200 students in a college. These students know at least one of three languages: English (E), Telugu (T), or Hindi (H). We are given the numbers of students knowing each language individually and in various combinations. Our goal is to determine how many students know only one of these three languages.

Analyzing the Given Data on Student Language Knowledge

Let's represent the number of students knowing each language or combination using set notation:

• Total students = 3200 (all know at least one language)
• Number of students knowing English, n(E) = 2400
• Number of students knowing Telugu, n(T) = 1700
• Number of students knowing Hindi, n(H) = 800
• Number of students knowing English and Telugu, n(E ∩ T) = 1000
• Number of students knowing English and Hindi, n(E ∩ H) = 500
• Number of students knowing Telugu and Hindi, n(T ∩ H) = 300
• Number of students knowing all three languages, n(E ∩ T ∩ H) = 100

Since all students know at least one language, the total number of students is equal to the number of students in the union of the three sets: n(E ∪ T ∪ H) = 3200.

Calculating Students Knowing Exactly Two Languages

To find the number of students who know only one language, we can first calculate the number of students who know exactly two languages. These are students who know two languages but *not* the third language. We subtract the number who know all three from each pair intersection:

• Students knowing English and Telugu only:
• $n(E \text{ and } T \text{ only}) = n(E \cap T) - n(E \cap T \cap H)$
• $n(E \text{ and } T \text{ only}) = 1000 - 100 = 900$
• Students knowing English and Hindi only:
• $n(E \text{ and } H \text{ only}) = n(E \cap H) - n(E \cap T \cap H)$
• $n(E \text{ and } H \text{ only}) = 500 - 100 = 400$
• Students knowing Telugu and Hindi only:
• $n(T \text{ and } H \text{ only}) = n(T \cap H) - n(E \cap T \cap H)$
• $n(T \text{ and } H \text{ only}) = 300 - 100 = 200$

The total number of students knowing exactly two languages is the sum of these values:

Number knowing exactly two languages = $900 + 400 + 200 = 1500$

Calculating Students Knowing Exactly Three Languages

The number of students knowing exactly three languages is directly given:

Number knowing exactly three languages = $n(E \cap T \cap H) = 100$

Finding Students Who Know Only One Language

The total number of students (3200) represents everyone who knows at least one language. This total is made up of students who know only one language, students who know exactly two languages, and students who know exactly three languages.

So, we can find the number of students who know only one language by subtracting the number who know exactly two or exactly three languages from the total number of students:

Number knowing only one language = Total students - Number knowing exactly two languages - Number knowing exactly three languages

Number knowing only one language = $3200 - 1500 - 100$

Number knowing only one language = $1700 - 100$

Number knowing only one language = $1600$

Alternatively, we could calculate the number knowing only one language for each language separately using the formula:

• $n(\text{only } E) = n(E) - [n(E \cap T) + n(E \cap H)] + n(E \cap T \cap H)$
• $n(\text{only } E) = 2400 - (1000 + 500) + 100 = 2400 - 1500 + 100 = 900 + 100 = 1000$
• $n(\text{only } T) = n(T) - [n(E \cap T) + n(T \cap H)] + n(E \cap T \cap H)$
• $n(\text{only } T) = 1700 - (1000 + 300) + 100 = 1700 - 1300 + 100 = 400 + 100 = 500$
• $n(\text{only } H) = n(H) - [n(E \cap H) + n(T \cap H)] + n(E \cap T \cap H)$
• $n(\text{only } H) = 800 - (500 + 300) + 100 = 800 - 800 + 100 = 0 + 100 = 100$

Total knowing only one language = $n(\text{only } E) + n(\text{only } T) + n(\text{only } H) = 1000 + 500 + 100 = 1600$.

Both methods yield the same result.

Conclusion

The number of students who know only one of the three languages is 1600.

Revision Table: College Language Problem

Additional Information: Set Theory and Venn Diagrams

This type of problem is a classic example that can be solved using set theory principles and visualized with a Venn diagram. A Venn diagram for three sets (English, Telugu, Hindi) would have 8 regions:

• The center region: Students knowing all three languages.
• Three regions for knowing exactly two languages (E&T only, E&H only, T&H only).
• Three regions for knowing exactly one language (E only, T only, H only).
• An outer region (outside the circles): Students knowing none of the languages (in this specific problem, this region is empty because all 3200 students know at least one).

The Principle of Inclusion-Exclusion helps us count elements in unions and intersections of sets accurately. The formula for the union of three sets used implicitly here is:

$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$

Knowing the values for the different intersections allows us to work outwards from the center of the Venn diagram to find the numbers in each specific region, including the 'only one' regions.