Out of 10000 people surveyed, 3700 liked city A, 4000 liked city B and 5000 liked city C. 700 people liked A and B 1200 liked B and C and 1000, liked A and C. Each person liked at least one city. Then find
The number of persons liking exactly two cities as a percentage of the number of people liking at least one city.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Sets
Let:
- \( A \) = set of people who like city A
- \( B \) = set of people who like city B
- \( C \) = set of people who like city C

Given data:
- \( n(A) = 3700 \)
- \( n(B) = 4000 \)
- \( n(C) = 5000 \)
- \( n(A \cap B) = 700 \) (people who like both A and B)
- \( n(B \cap C) = 1200 \) (people who like both B and C)
- \( n(A \cap C) = 1000 \) (people who like both A and C)

### Step 2: Use Inclusion-Exclusion Principle
We need to find the number of people who like all three cities, denoted as \( n(A \cap B \cap C) \).

Using the inclusion-exclusion principle, we can express the total number of people who like at least one city as follows:
\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) \]

### Step 3: Set Up the Equation
Substituting the known values into the equation:
\[ n(A \cup B \cup C) = 3700 + 4000 + 5000 - 700 - 1200 - 1000 + n(A \cap B \cap C) \]
\[ n(A \cup B \cup C) = 12700 - 2900 + n(A \cap B \cap C) \]
\[ n(A \cup B \cup C) = 9800 + n(A \cap B \cap C) \]

Since we know that the total number of surveyed people is 10000, we can set up the equation:
\[ 10000 = 9800 + n(A \cap B \cap C) \]
This simplifies to:
\[ n(A \cap B \cap C) = 200 \]

### Step 4: Find the Number of People Liking Exactly Two Cities
Now, we can find the number of people who like exactly two cities:
- People who like A and B but not C:
\[ n(A \cap B) - n(A \cap B \cap C) = 700 - 200 = 500 \]
- People who like B and C but not A:
\[ n(B \cap C) - n(A \cap B \cap C) = 1200 - 200 = 1000 \]
- People who like A and C but not B:
\[ n(A \cap C) - n(A \cap B \cap C) = 1000 - 200 = 800 \]

Adding these gives the total number of people who like exactly two cities:
\[ 500 + 1000 + 800 = 2300 \]

### Step 5: Calculate the Percentage
Now, we need to find the percentage of people liking exactly two cities out of those who like at least one city:
\[ \text{Percentage} = \left( \frac{\text{Number of people liking exactly 2 cities}}{\text{Total number of people}} \right) \times 100 \]
\[ \text{Percentage} = \left( \frac{2300}{10000} \right) \times 100 = 23\% \]

### Final Answer
The number of persons liking exactly two cities as a percentage of the number of people liking at least one city is **23%**.

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