A hall has 12 gates. In how many ways, can a man enter the hall through one gate and come out through a different gate?

Answer

To solve the problem of how many ways a man can enter a hall through one gate and come out through a different gate when there are 12 gates, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of Gates**:
The total number of gates available is 12.

2. **Choose the Gate to Enter**:
The man can choose any one of the 12 gates to enter the hall. Therefore, there are 12 options for the entrance.

3. **Choose the Gate to Exit**:
After entering through one gate, the man must exit through a different gate. Since he has already entered through one gate, there are 11 gates left to choose from for exiting.

4. **Calculate the Total Combinations**:
To find the total number of ways to enter and exit, we multiply the number of choices for entering by the number of choices for exiting:
\[ \text{Total Ways} = (\text{Number of Gates to Enter}) \times (\text{Number of Gates to Exit}) = 12 \times 11 \]

5. **Perform the Calculation**:
\[ 12 \times 11 = 132 \]

Thus, the total number of ways the man can enter the hall through one gate and exit through a different gate is **132**.