in a class of 35 students , 17 have taken Mathematics , 10 have taken mathematics but not economics if each student has taken erither mathematics Of Economics or both , then the number of students who have taken Economics but not mathematics is

Options

A. 7
B. 25
C. 18
D. 32

Answer

To solve the problem step by step, we will use the information provided and apply set theory concepts.

### Step 1: Identify the given information
- Total number of students in the class (N) = 35
- Number of students who have taken Mathematics (n(M)) = 17
- Number of students who have taken Mathematics but not Economics (n(M - E)) = 10

### Step 2: Use the relationship between sets
From the information given, we can determine the number of students who have taken both Mathematics and Economics (n(M ∩ E)).

Using the formula:
\[ n(M) = n(M - E) + n(M ∩ E) \]
Substituting the known values:
\[ 17 = 10 + n(M ∩ E) \]

### Step 3: Solve for n(M ∩ E)
Rearranging the equation:
\[ n(M ∩ E) = 17 - 10 \]
\[ n(M ∩ E) = 7 \]

### Step 4: Find the number of students who have taken Economics (n(E))
We know that all students have taken either Mathematics, Economics, or both. Therefore, we can use the formula for the union of two sets:
\[ n(M ∪ E) = n(M) + n(E) - n(M ∩ E) \]
Given that \( n(M ∪ E) = 35 \):
\[ 35 = 17 + n(E) - 7 \]

### Step 5: Solve for n(E)
Rearranging the equation:
\[ n(E) = 35 - 17 + 7 \]
\[ n(E) = 25 \]

### Step 6: Find the number of students who have taken Economics but not Mathematics (n(E - M))
Using the formula:
\[ n(E - M) = n(E) - n(M ∩ E) \]
Substituting the known values:
\[ n(E - M) = 25 - 7 \]
\[ n(E - M) = 18 \]

### Final Answer
The number of students who have taken Economics but not Mathematics is **18**.

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