In a group of 70 persons, 37 like coffee and 52 like tea. Each person like atleast one drink. Find how many persons like both drink ?

Answer

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

### Step 1: Understand the problem
We have a total of 70 persons. Out of these:
- 37 persons like coffee.
- 52 persons like tea.
- Each person likes at least one of the two drinks.

We need to find out how many persons like both coffee and tea.

### Step 2: Use the formula for the union of two sets
The formula for the number of elements in the union of two sets is given by:
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
Where:
- \(|A \cup B|\) is the number of elements in the union of sets A and B (people who like at least one drink).
- \(|A|\) is the number of elements in set A (people who like coffee).
- \(|B|\) is the number of elements in set B (people who like tea).
- \(|A \cap B|\) is the number of elements in the intersection of sets A and B (people who like both drinks).

### Step 3: Substitute the known values
From the problem:
- \(|A| = 37\) (people who like coffee)
- \(|B| = 52\) (people who like tea)
- \(|A \cup B| = 70\) (total people)

Now substituting these values into the formula:
\[ 70 = 37 + 52 - |A \cap B| \]

### Step 4: Simplify the equation
Combine the numbers on the right side:
\[ 70 = 89 - |A \cap B| \]

### Step 5: Solve for \(|A \cap B|\)
Rearranging the equation gives:
\[ |A \cap B| = 89 - 70 \]
\[ |A \cap B| = 19 \]

### Conclusion
Thus, the number of persons who like both coffee and tea is **19**.

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