An investigator interviewed 100 students to determine the performance of three drinks: milk , coffee and tea the in vestigotor reported that 10 students take all three drinks milk . Coffe and tea , 20 students take take coffe , 25 students take milk only , 5 students take coffee only and 8 students take tea only . then the number of students who did not take any of these drinks is

Options

A. 10
B. 20
C. 25
D. 30

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion and a Venn diagram approach to visualize the distribution of students among the three drinks: milk, coffee, and tea.

### Step-by-Step Solution:

1. **Identify the given data:**
- Total students interviewed: \(100\)
- Students taking all three drinks (milk, coffee, tea): \(10\)
- Students taking coffee and milk (including those who take all three): \(20\)
- Students taking milk only: \(25\)
- Students taking coffee only: \(5\)
- Students taking tea only: \(8\)

2. **Determine the number of students who take both milk and coffee but not tea:**
- Students taking milk and coffee (20) includes those who take all three drinks (10).
- Therefore, students taking only milk and coffee = \(20 - 10 = 10\).

3. **Determine the number of students who take both milk and tea but not coffee:**
- Let’s denote the number of students taking milk and tea (including those who take all three) as \(x\).
- We know that \(x = 25\) (total taking milk and tea).
- Therefore, students taking only milk and tea = \(25 - 10 = 15\).

4. **Determine the number of students who take both coffee and tea but not milk:**
- Let’s denote the number of students taking coffee and tea (including those who take all three) as \(y\).
- We know that \(y = 30\) (total taking coffee and tea).
- Therefore, students taking only coffee and tea = \(30 - 10 = 20\).

5. **Summarize the distribution of students:**
- Students taking only milk: \(25\)
- Students taking only coffee: \(5\)
- Students taking only tea: \(8\)
- Students taking only milk and coffee: \(10\)
- Students taking only milk and tea: \(15\)
- Students taking only coffee and tea: \(20\)
- Students taking all three drinks: \(10\)

6. **Calculate the total number of students who took at least one drink:**
- Total = Students taking only milk + Students taking only coffee + Students taking only tea + Students taking only milk and coffee + Students taking only milk and tea + Students taking only coffee and tea + Students taking all three drinks
- Total = \(25 + 5 + 8 + 10 + 15 + 20 + 10 = 93\)

7. **Calculate the number of students who did not take any drink:**
- Students who did not take any drink = Total students interviewed - Total students who took at least one drink
- Students who did not take any drink = \(100 - 93 = 7\)

### Final Answer:
The number of students who did not take any of these drinks is **7**.