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 A and C and 1000, liked B and C. Each person liked at least one city. Then find
The number of persons liking A and B but not C.

Answer

To solve the problem step by step, we will use the Venn Diagram method to visualize the relationships between the people who like the different cities.

### Step 1: Define the Variables
Let:
- \( A \) = Number of people who like City A = 3700
- \( B \) = Number of people who like City B = 4000
- \( C \) = Number of people who like City C = 5000
- \( AB \) = Number of people who like both A and B = 700
- \( AC \) = Number of people who like both A and C = 1200
- \( BC \) = Number of people who like both B and C = 1000
- \( ABC \) = Number of people who like all three cities (A, B, and C) = \( x \)

### Step 2: Set Up the Equations
From the information given, we can express the number of people who like only two cities:
- People who like A and B but not C = \( AB - x = 700 - x \)
- People who like A and C but not B = \( AC - x = 1200 - x \)
- People who like B and C but not A = \( BC - x = 1000 - x \)

### Step 3: Total People Calculation
The total number of people surveyed is 10,000. Since each person likes at least one city, we can set up the equation:
\[ (A - (AB - x) - (AC - x) - x) + (B - (AB - x) - (BC - x) - x) + (C - (AC - x) - (BC - x) - x) + (AB - x) + (AC - x) + (BC - x) + x = 10000 \]

### Step 4: Substitute Values
Now substituting the values we have:
\[ (3700 - (700 - x) - (1200 - x) - x) + (4000 - (700 - x) - (1000 - x) - x) + (5000 - (1200 - x) - (1000 - x) - x) + (700 - x) + (1200 - x) + (1000 - x) + x = 10000 \]

### Step 5: Simplify the Equation
This simplifies to:
\[ 3700 - 700 + x - 1200 + x - x + 4000 - 700 + x - 1000 + x - x + 5000 - 1200 + x - 1000 + x - x + 700 - x + 1200 - x + 1000 - x + x = 10000 \]

Combining like terms, we get:
\[ 3700 + 4000 + 5000 - 2900 + 3x = 10000 \]
\[ 10000 + 3x = 10000 \]
\[ 3x = 2700 \]
\[ x = 200 \]

### Step 6: Find the Number of Persons Liking A and B but Not C
Now that we have \( x = 200 \), we can find the number of people who like A and B but not C:
\[ AB - x = 700 - 200 = 500 \]

### Final Answer
The number of persons liking A and B but not C is **500**.

---