There is a group of 50 persons who can speak English or Tamil or both. Out of these persons , 37 can speak English and 30 cm speak Tamil.
How many can speak both English and Tamil

Answer

To find out how many persons can speak both English and Tamil, we can use the principle of inclusion-exclusion. Here’s a step-by-step solution:

### Step 1: Define the sets
Let:
- \( A \) = the set of persons who can speak English
- \( B \) = the set of persons who can speak Tamil

### Step 2: Identify the given values
From the problem, we know:
- Total number of persons, \( N(A \cup B) = 50 \)
- Number of persons who can speak English, \( N(A) = 37 \)
- Number of persons who can speak Tamil, \( N(B) = 30 \)

### Step 3: Use the inclusion-exclusion principle
The formula for the union of two sets is given by:
\[ N(A \cup B) = N(A) + N(B) - N(A \cap B) \]
Where \( N(A \cap B) \) is the number of persons who can speak both English and Tamil.

### Step 4: Substitute the known values into the formula
Substituting the known values into the formula:
\[ 50 = 37 + 30 - N(A \cap B) \]

### Step 5: Simplify the equation
Now, simplify the equation:
\[ 50 = 67 - N(A \cap B) \]

### Step 6: Solve for \( N(A \cap B) \)
Rearranging the equation gives:
\[ N(A \cap B) = 67 - 50 \]
\[ N(A \cap B) = 17 \]

### Conclusion
Thus, the number of persons who can speak both English and Tamil is **17**.

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