Out of 20 members in a family , 11 like to take tea and 14 like coffee . Assume that each one likes alteast one of the two drinks . How many like :
(i) both tea and coffee
(ii) only tea and not coffee
(iii) only coffee and not tea ?

Answer

To solve the problem step by step, we will use the principle of set theory and Venn diagrams.

### Step 1: Define the Sets
Let:
- \( T \) = the set of members who like tea
- \( C \) = the set of members who like coffee

From the problem, we know:
- Total members in the family, \( |U| = 20 \)
- Members who like tea, \( |T| = 11 \)
- Members who like coffee, \( |C| = 14 \)

### Step 2: Use the Union Formula
Since every member likes at least one of the two drinks, we can use the formula for the union of two sets:
\[ |T \cup C| = |T| + |C| - |T \cap C| \]
Where:
- \( |T \cup C| \) is the number of members who like either tea or coffee (or both).
- \( |T \cap C| \) is the number of members who like both tea and coffee.

### Step 3: Substitute Known Values
We know that \( |T \cup C| = 20 \) (since all members like at least one drink). Substituting the known values into the formula gives:
\[ 20 = 11 + 14 - |T \cap C| \]

### Step 4: Solve for \( |T \cap C| \)
Now, we can solve for \( |T \cap C| \):
\[ 20 = 25 - |T \cap C| \]
\[ |T \cap C| = 25 - 20 = 5 \]
So, \( |T \cap C| = 5 \). This means 5 members like both tea and coffee.

### Step 5: Find Members Who Like Only Tea
To find the number of members who like only tea (not coffee), we subtract those who like both from those who like tea:
\[ |T \text{ only}| = |T| - |T \cap C| = 11 - 5 = 6 \]

### Step 6: Find Members Who Like Only Coffee
Similarly, to find the number of members who like only coffee (not tea), we subtract those who like both from those who like coffee:
\[ |C \text{ only}| = |C| - |T \cap C| = 14 - 5 = 9 \]

### Final Answers
Now we can summarize the results:
1. Number of members who like both tea and coffee: \( |T \cap C| = 5 \)
2. Number of members who like only tea and not coffee: \( |T \text{ only}| = 6 \)
3. Number of members who like only coffee and not tea: \( |C \text{ only}| = 9 \)