An lnvestigator interviewed `100` students to determine the performance of three drinks milk, coffee and tea. The investigator reported that `10` students take all three drinks, 20 students take milk and coffee, `30` students take coffee and tea, `25` students take milk and tea, `12` studnts take milk only. `5` students take coffee only and `8` student take tea only. Then the number of students who did not take any of the three drinks is `lambda` . The value of `(lambda)/(4)` is equal to

Options

A. `2`
B. `3`
C. `4`
D. `5`

Explanation

To solve the problem step by step, we will use a Venn diagram to organize the information given about the students and their drink preferences.

### Step 1: Define the Variables
Let:
- \( M \): Students who drink milk
- \( C \): Students who drink coffee
- \( T \): Students who drink tea

### Step 2: Write Down the Given Information
From the problem, we have the following data:
- \( |M \cap C \cap T| = 10 \) (students who drink all three)
- \( |M \cap C| = 20 \) (students who drink milk and coffee)
- \( |C \cap T| = 30 \) (students who drink coffee and tea)
- \( |M \cap T| = 25 \) (students who drink milk and tea)
- \( |M| = 12 \) (students who drink only milk)
- \( |C| = 5 \) (students who drink only coffee)
- \( |T| = 8 \) (students who drink only tea)

### Step 3: Fill in the Venn Diagram
1. **Students who drink only milk**: \( 12 \)
2. **Students who drink only coffee**: \( 5 \)
3. **Students who drink only tea**: \( 8 \)
4. **Students who drink all three drinks**: \( 10 \)

Now, we need to find the number of students who drink exactly two types of drinks:
- For milk and coffee:
\[ |M \cap C| - |M \cap C \cap T| = 20 - 10 = 10 \]
- For coffee and tea:
\[ |C \cap T| - |M \cap C \cap T| = 30 - 10 = 20 \]
- For milk and tea:
\[ |M \cap T| - |M \cap C \cap T| = 25 - 10 = 15 \]

### Step 4: Summarize the Venn Diagram
Now we can summarize the counts:
- Milk only: \( 12 \)
- Coffee only: \( 5 \)
- Tea only: \( 8 \)
- Milk and coffee only: \( 10 \)
- Coffee and tea only: \( 20 \)
- Milk and tea only: \( 15 \)
- All three drinks: \( 10 \)

### Step 5: Calculate the Total Number of Students Who Drink
Now we will add all these values to find the total number of students who drink at least one type of drink:
\[ \text{Total} = 12 + 5 + 8 + 10 + 20 + 15 + 10 = 80 \]

### Step 6: Calculate the Number of Students Who Did Not Drink Any
The total number of students surveyed is \( 100 \). Therefore, the number of students who did not take any of the three drinks (denote this as \( \lambda \)) is:
\[ \lambda = 100 - 80 = 20 \]

### Step 7: Calculate \( \frac{\lambda}{4} \)
Now, we need to find \( \frac{\lambda}{4} \):
\[ \frac{\lambda}{4} = \frac{20}{4} = 5 \]

### Final Answer
The value of \( \frac{\lambda}{4} \) is \( 5 \).

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