In a group of 70 people, 48 speak Tamil, 36 speak English and all the people speak at least one language. Find How many speak both the languages ?

Answer

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

### Step-by-Step Solution:

1. **Identify the Sets**:
- Let \( A \) be the set of people who speak Tamil.
- Let \( B \) be the set of people who speak English.
- We are given:
- \( n(A) = 48 \) (number of people who speak Tamil)
- \( n(B) = 36 \) (number of people who speak English)
- \( n(A \cup B) = 70 \) (total number of people who speak at least one language)

2. **Use the Inclusion-Exclusion Principle**:
- According to the principle, the total number of people who speak at least one language can be expressed as:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
- Here, \( n(A \cap B) \) represents the number of people who speak both languages.

3. **Substitute the Known Values**:
- Substitute the values we have into the equation:
\[ 70 = 48 + 36 - n(A \cap B) \]

4. **Simplify the Equation**:
- First, calculate \( 48 + 36 \):
\[ 48 + 36 = 84 \]
- Now substitute back into the equation:
\[ 70 = 84 - n(A \cap B) \]

5. **Solve for \( n(A \cap B) \)**:
- Rearranging the equation gives:
\[ n(A \cap B) = 84 - 70 \]
- Therefore:
\[ n(A \cap B) = 14 \]

6. **Conclusion**:
- The number of people who speak both languages is \( 14 \).

### Final Answer:
The number of people who speak both Tamil and English is **14**.