In a group of 52 persons, 16 drink tea but not coffee and 33 drink tea. Then, the number of persons who drink coffee but not tea, is

Options

A. 18
B. 19
C. 20
D. 21

Answer

To solve the problem step by step, we will use the concept of sets and the principle of inclusion-exclusion.

### Step 1: Define the Sets
Let:
- \( A \) = the set of persons who drink tea
- \( B \) = the set of persons who drink coffee

From the problem, we have:
- Total number of persons, \( n(U) = 52 \)
- Number of persons who drink tea, \( n(A) = 33 \)
- Number of persons who drink tea but not coffee, \( n(A - B) = 16 \)

### Step 2: Find the Number of Persons Who Drink Both Tea and Coffee
We can find the number of persons who drink both tea and coffee using the formula:
\[ n(A) = n(A - B) + n(A \cap B) \]
Substituting the known values:
\[ 33 = 16 + n(A \cap B) \]
Solving for \( n(A \cap B) \):
\[ n(A \cap B) = 33 - 16 = 17 \]

### Step 3: Use the Inclusion-Exclusion Principle
Now, we can find the total number of persons who drink either tea or coffee (or both) using the inclusion-exclusion principle:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
We know \( n(A) = 33 \) and \( n(A \cap B) = 17 \). We need to find \( n(B) \) (the total number of coffee drinkers).

### Step 4: Find the Total Number of Coffee Drinkers
Since we know the total number of persons is 52, we can express \( n(B) \) as:
\[ n(U) = n(A \cup B) + n(B - A) \]
Where \( n(B - A) \) is the number of persons who drink coffee but not tea. Rearranging gives:
\[ n(B) = n(U) - n(A \cup B) + n(B - A) \]

### Step 5: Calculate \( n(B) \)
We need to find \( n(B) \) first. We can express \( n(B) \) as:
\[ n(B) = n(A \cap B) + n(B - A) \]
Let \( n(B - A) \) be the number of persons who drink coffee but not tea. Then:
\[ 52 = n(A) + n(B) - n(A \cap B) \]
Substituting the known values:
\[ 52 = 33 + n(B) - 17 \]
This simplifies to:
\[ 52 = 16 + n(B) \]
So,
\[ n(B) = 52 - 16 = 36 \]

### Step 6: Find the Number of Persons Who Drink Coffee but Not Tea
Now we can find \( n(B - A) \):
\[ n(B) = n(A \cap B) + n(B - A) \]
Substituting the known values:
\[ 36 = 17 + n(B - A) \]
Solving for \( n(B - A) \):
\[ n(B - A) = 36 - 17 = 19 \]

### Final Answer
The number of persons who drink coffee but not tea is **19**.

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