There are three mathematics teachers in a college in which there are 6 classes. In how many different ways can they choose the classes provided one teaches one class only

Answer

To solve the problem of how many different ways three mathematics teachers can choose classes from six available classes, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number of teachers and classes**:
- We have 3 mathematics teachers.
- We have 6 classes available.

2. **Understand the selection process**:
- Each teacher will choose one class, and no class can be taught by more than one teacher. This means that the selection is sequential.

3. **Calculate the choices for each teacher**:
- The first teacher can choose any of the 6 classes. This gives us \( 6C1 \) (which is simply 6).
- After the first teacher has chosen a class, there are 5 classes left for the second teacher to choose from. This gives us \( 5C1 \) (which is 5).
- Finally, the third teacher will choose from the remaining 4 classes. This gives us \( 4C1 \) (which is 4).

4. **Multiply the choices**:
- To find the total number of ways the teachers can choose the classes, we multiply the number of choices for each teacher:
\[ \text{Total ways} = 6C1 \times 5C1 \times 4C1 = 6 \times 5 \times 4 \]

5. **Calculate the final result**:
- Now, we perform the multiplication:
\[ 6 \times 5 = 30 \]
\[ 30 \times 4 = 120 \]

6. **Conclusion**:
- Therefore, the total number of different ways the three mathematics teachers can choose the classes is **120**.