The Students of a class are offered three language (Hindi. English and French). 15 students learn all the three language whereas 28 students do not learn any language. The number of students learning Hindi and English but not French is twice the number of students learning Hindi and French but not English. The number of students learning English and French but not Hindi is thrice the number of students learning Hindi and French but not English. 23 students learn only Hindi and 17 students learn only English. The total number of students learning French is 46 and the total number of students learning only French is 11.
How many students learn at least one language?

Options

A. 45
B. 51
C. 96
D. None of these

Answer

To solve the problem step by step, we will use the information given in the question and organize it systematically.

### Step 1: Define Variables
Let:
- \( x \) = Number of students learning Hindi and French but not English.
- \( 2x \) = Number of students learning Hindi and English but not French (twice the number of students learning Hindi and French).
- \( 3x \) = Number of students learning English and French but not Hindi (thrice the number of students learning Hindi and French).

### Step 2: Gather Given Information
From the problem, we know:
- 15 students learn all three languages (Hindi, English, and French).
- 28 students do not learn any language.
- 23 students learn only Hindi.
- 17 students learn only English.
- 11 students learn only French.
- Total number of students learning French = 46.

### Step 3: Set Up the Total for French Learners
The total number of students learning French can be expressed as:
\[ \text{Total French learners} = \text{Only French} + \text{Hindi and French} + \text{English and French} + \text{All three} \]
Substituting the known values:
\[ 46 = 11 + x + 3x + 15 \]
This simplifies to:
\[ 46 = 26 + 4x \]
\[ 20 = 4x \implies x = 5 \]

### Step 4: Calculate Other Groups
Now we can calculate the number of students in each group:
- Students learning Hindi and French but not English = \( x = 5 \).
- Students learning Hindi and English but not French = \( 2x = 10 \).
- Students learning English and French but not Hindi = \( 3x = 15 \).

### Step 5: Total Students Learning Each Language
Now we can summarize the total number of students learning each language:
- Only Hindi = 23
- Only English = 17
- Only French = 11
- Hindi and French but not English = 5
- Hindi and English but not French = 10
- English and French but not Hindi = 15
- All three languages = 15

### Step 6: Calculate Total Students Learning at Least One Language
To find the total number of students learning at least one language, we add all the groups:
\[ \text{Total} = \text{Only Hindi} + \text{Only English} + \text{Only French} + \text{Hindi and French} + \text{Hindi and English} + \text{English and French} + \text{All three} \]
Substituting the values:
\[ \text{Total} = 23 + 17 + 11 + 5 + 10 + 15 + 15 = 96 \]

### Step 7: Account for Students Not Learning Any Language
Since 28 students do not learn any language, we need to add them to our total:
\[ \text{Total students in class} = 96 + 28 = 124 \]

### Final Answer
Thus, the number of students learning at least one language is:
\[ \boxed{96} \]