In a group of 36 persons ,20 take coffee but not tea. 16 take tea coffee .Find the number of persons who take niether tea nor coffee.

Options

A. 2
B. 1
C. 0
D. 3

Explanation

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

### Step 1: Understand the given information
We have a total of 36 persons in the group.
- 20 persons take coffee but not tea.
- 16 persons take both tea and coffee.

### Step 2: Define the sets
Let:
- \( C \) = the number of persons who take coffee
- \( T \) = the number of persons who take tea
- \( N \) = the number of persons who take neither tea nor coffee

### Step 3: Calculate the total number of persons who take coffee
From the information given:
- Persons who take only coffee = 20
- Persons who take both coffee and tea = 16

Thus, the total number of persons who take coffee \( C \) is:
\[ C = \text{(only coffee)} + \text{(both coffee and tea)} = 20 + 16 = 36 \]

### Step 4: Calculate the total number of persons who take tea
Since we know that 16 persons take both coffee and tea, we need to find out how many take only tea. However, we don't have that information directly. We can denote the number of persons who take only tea as \( x \).

So, the total number of persons who take tea \( T \) is:
\[ T = x + 16 \]

### Step 5: Use the total number of persons
The total number of persons in the group is 36. Therefore, we can express this as:
\[ \text{(only coffee)} + \text{(only tea)} + \text{(both)} + \text{(neither)} = 36 \]
Substituting the known values:
\[ 20 + x + 16 + N = 36 \]

### Step 6: Simplify the equation
Combining the known values:
\[ 36 + x + N = 36 \]

### Step 7: Solve for \( N \)
Rearranging the equation gives us:
\[ x + N = 0 \]
This implies:
\[ N = -x \]

Since \( N \) (the number of persons who take neither tea nor coffee) cannot be negative, this means that:
\[ x = 0 \]

### Step 8: Conclusion
Since \( x = 0 \), it means there are no persons who take only tea. Therefore, all 36 persons are accounted for by those who take coffee (20 who take only coffee and 16 who take both). Thus, the number of persons who take neither tea nor coffee is:
\[ N = 0 \]

### Final Answer
The number of persons who take neither tea nor coffee is **0**.

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