In a group of 25 people, 17 drink tea. Out of these 10 drink only tea. Find how many people drink coffee but not tea ? It is given that each one takes atleast one drink.

Answer

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

### Step 1: Define the Variables
Let:
- \( T \) = Number of people who drink tea
- \( C \) = Number of people who drink coffee
- \( T \cap C \) = Number of people who drink both tea and coffee
- \( T - C \) = Number of people who drink only tea
- \( C - T \) = Number of people who drink only coffee

From the problem, we know:
- Total number of people = 25
- Number of people who drink tea (\( T \)) = 17
- Number of people who drink only tea (\( T - C \)) = 10

### Step 2: Calculate the Number of People Who Drink Both Tea and Coffee
Since 10 people drink only tea, we can find the number of people who drink both tea and coffee (\( T \cap C \)):
\[ T = (T - C) + (T \cap C) \]
Substituting the known values:
\[ 17 = 10 + (T \cap C) \]
Solving for \( T \cap C \):
\[ T \cap C = 17 - 10 = 7 \]

### Step 3: Calculate the Number of People Who Drink Coffee
Using the total number of people, we can find the number of people who drink coffee (\( C \)):
\[ n(T \cup C) = n(T) + n(C) - n(T \cap C) \]
We know:
- \( n(T \cup C) = 25 \)
- \( n(T) = 17 \)
- \( n(T \cap C) = 7 \)

Substituting these values into the formula:
\[ 25 = 17 + n(C) - 7 \]
Simplifying:
\[ 25 = 10 + n(C) \]
Thus,
\[ n(C) = 25 - 10 = 15 \]

### Step 4: Calculate the Number of People Who Drink Only Coffee
Now, we can find the number of people who drink only coffee (\( C - T \)):
\[ C = (C - T) + (T \cap C) \]
Substituting the known values:
\[ 15 = (C - T) + 7 \]
Solving for \( C - T \):
\[ C - T = 15 - 7 = 8 \]

### Final Answer
The number of people who drink coffee but not tea is \( 8 \).

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