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 people liking all the three cities.

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion for three sets. Let's denote the number of people who liked city A, B, and C as follows:

- Let \( n(A) = 3700 \) (people who liked city A)
- Let \( n(B) = 4000 \) (people who liked city B)
- Let \( n(C) = 5000 \) (people who liked city C)
- Let \( n(A \cap B) = 700 \) (people who liked both A and B)
- Let \( n(B \cap C) = 1200 \) (people who liked both B and C)
- Let \( n(A \cap C) = 1000 \) (people who liked both A and C)

We need to find the number of people who liked all three cities, denoted as \( n(A \cap B \cap C) \).

### Step 1: Use the formula for the union of three sets

The formula for the union of three sets is given by:

\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) \]

### Step 2: Substitute the known values

We know that the total number of people surveyed is 10,000, which means:

\[ n(A \cup B \cup C) = 10000 \]

Now substituting the known values into the formula:

\[ 10000 = 3700 + 4000 + 5000 - 700 - 1200 - 1000 + n(A \cap B \cap C) \]

### Step 3: Simplify the equation

Calculating the sum and differences:

\[ 10000 = 3700 + 4000 + 5000 = 12700 \]
\[ 12700 - 700 - 1200 - 1000 = 12700 - 2900 = 9800 \]

So we have:

\[ 10000 = 9800 + n(A \cap B \cap C) \]

### Step 4: Solve for \( n(A \cap B \cap C) \)

Now, we can isolate \( n(A \cap B \cap C) \):

\[ n(A \cap B \cap C) = 10000 - 9800 = 200 \]

### Conclusion

Thus, the number of people who liked all three cities A, B, and C is:

\[ \boxed{200} \]