Understanding Language Knowledge in College

This problem provides data about the language skills of 3200 students in a college. Every student knows at least one of three languages: English, Telugu, or Hindi. We are given the total number of students who know each language individually, the number who know specific pairs of languages, and the number who know all three languages. The goal is to find the number of students who know only Hindi.

Given Information on Language Speakers

Let's list the provided data points clearly:

• Total number of students = 3200
• Number of students who know English = 2400
• Number of students who know Telugu = 1700
• Number of students who know Hindi = 800
• Number of students who know English and Telugu = 1000
• Number of students who know English and Hindi = 500
• Number of students who know Telugu and Hindi = 300
• Number of students who know English, Telugu, and Hindi = 100

All 3200 students know at least one language, meaning the union of the three sets of students (English speakers, Telugu speakers, Hindi speakers) is equal to the total number of students.

Calculating Students Who Know Only Hindi

To find the number of students who know only Hindi, we need to take the total number of students who know Hindi and subtract those who also know other languages. The group of students who know Hindi can be divided into four parts:

1. Students who know only Hindi.
2. Students who know Hindi and English, but not Telugu (Hindi & English only).
3. Students who know Hindi and Telugu, but not English (Hindi & Telugu only).
4. Students who know English, Telugu, and Hindi (all three).

We are given the numbers for the total who know Hindi, those who know Hindi and English (which includes those who know all three), those who know Hindi and Telugu (which includes those who know all three), and those who know all three.

Let H be the set of students who know Hindi, E be the set who know English, and T be the set who know Telugu.

• Total who know Hindi, $|H| = 800$
• Total who know Hindi and English, $|H \cap E| = 500$
• Total who know Hindi and Telugu, $|H \cap T| = 300$
• Total who know English, Telugu, and Hindi, $|H \cap E \cap T| = 100$

The students who know Hindi and English ONLY (not Telugu) are those in $|H \cap E|$ excluding those who also know Telugu. This is given by:

Students knowing Hindi and English only = $|H \cap E| - |H \cap E \cap T|$

\text{Hindi & English only} = 500 - 100 = 400

The students who know Hindi and Telugu ONLY (not English) are those in $|H \cap T|$ excluding those who also know English. This is given by:

Students knowing Hindi and Telugu only = $|H \cap T| - |H \cap E \cap T|$

\text{Hindi & Telugu only} = 300 - 100 = 200

Now, the total number of students who know Hindi is the sum of those knowing only Hindi, those knowing Hindi and English only, those knowing Hindi and Telugu only, and those knowing all three languages.

$|H| = (\text{only Hindi}) + (\text{Hindi & English only}) + (\text{Hindi & Telugu only}) + (|H \cap E \cap T|)$

Substitute the known values:

800 = (\text{only Hindi}) + 400 + 200 + 100

800 = (\text{only Hindi}) + 700

To find the number of students who know only Hindi, subtract 700 from 800:

\text{only Hindi} = 800 - 700 = 100

Therefore, 100 students know only Hindi.

Summary of Results

We can summarize the number of students in each specific language group:

• Only English: Calculated similarly using $|E|$, $|E \cap T|$, $|E \cap H|$, $|E \cap T \cap H|$
• Only Telugu: Calculated similarly using $|T|$, $|E \cap T|$, $|T \cap H|$, $|E \cap T \cap H|$
• Only Hindi: 100 (as calculated above)
• English and Telugu only: $|E \cap T| - |E \cap T \cap H| = 1000 - 100 = 900$
• English and Hindi only: $|E \cap H| - |E \cap T \cap H| = 500 - 100 = 400$
• Telugu and Hindi only: $|T \cap H| - |E \cap T \cap H| = 300 - 100 = 200$
• English, Telugu, and Hindi: 100

Let's verify the total:

\text{Total students} = (\text{Only E}) + (\text{Only T}) + (\text{Only H}) + (\text{E & T only}) + (\text{E & H only}) + (\text{T & H only}) + (\text{E & T & H})

We need Only E and Only T first:

Only E = $|E| - (|E \cap T| - |E \cap T \cap H|) - (|E \cap H| - |E \cap T \cap H|) - |E \cap T \cap H|$

Only E = $2400 - (1000 - 100) - (500 - 100) - 100$

Only E = $2400 - 900 - 400 - 100 = 2400 - 1400 = 1000$

Only T = $|T| - (|E \cap T| - |E \cap T \cap H|) - (|T \cap H| - |E \cap T \cap H|) - |E \cap T \cap H|$

Only T = $1700 - (1000 - 100) - (300 - 100) - 100$

Only T = $1700 - 900 - 200 - 100 = 1700 - 1200 = 500$

Now sum all the calculated segments:

\text{Total} = 1000 (\text{Only E}) + 500 (\text{Only T}) + 100 (\text{Only H}) + 900 (\text{E & T only}) + 400 (\text{E & H only}) + 200 (\text{T & H only}) + 100 (\text{All three})

\text{Total} = 1000 + 500 + 100 + 900 + 400 + 200 + 100 = 3200

This matches the total number of students given, confirming our calculations for each segment are correct.

Conclusion

Based on the calculations, the number of students who know only Hindi is 100.

Revision Table: Language Knowledge Statistics

Additional Information: Set Theory and Venn Diagrams

This type of problem involving overlapping groups is commonly solved using set theory principles and visualized with Venn diagrams. A Venn diagram for three sets (English, Telugu, Hindi) would have 8 distinct regions, representing:

• Only English
• Only Telugu
• Only Hindi
• English and Telugu only
• English and Hindi only
• Telugu and Hindi only
• English, Telugu, and Hindi (the center intersection)
• None of the languages (outside the circles)

In this specific problem, we are told all 3200 students know at least one language, meaning the region outside the three circles is 0. The principle of inclusion-exclusion is a formula that relates the sizes of unions and intersections of sets:

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

This formula was used to verify the consistency of the given data.

To find the number of elements in a region representing "only one set" (like only Hindi), you take the total in that set and subtract the overlaps, being careful to add back the innermost intersection (all three) because it was subtracted multiple times.

For example, Only H = $|H|$ - (those in H and E) - (those in H and T) + (those in H, E, and T).

This corresponds to: Only H = $|H| - (|H \cap E| + |H \cap T|) + |H \cap E \cap T|$

Using the numbers: Only H = $800 - (500 + 300) + 100 = 800 - 800 + 100 = 100$.

This provides an alternative calculation method using the direct inclusion-exclusion concept for 'only H', which confirms our earlier result obtained by breaking down the 'Total H' group.