To solve the problem, we calculate the number of ways to arrange the train departures for the week, based on the given constraints.
### Step-by-step Solution:
1. **Understand the problem**:
- There are 7 days in a week.
- Train schedules:
- **Three days:** Two trains depart twice a day.
- **Two days:** One train departs daily.
- **Two days:** Three trains depart once a day.
- We need to arrange these 7 days into groups satisfying the above conditions.
2. **Label the days for clarity**:
- Let "A" represent the days with two trains twice a day (3 days total).
- Let "B" represent the days with one train daily (2 days total).
- Let "C" represent the days with three trains once a day (2 days total).
3. **Determine the total number of days to arrange**:
There are 7 days in total:
- 3 days of type A,
- 2 days of type B,
- 2 days of type C.
4. **Apply the permutation formula**:
The number of ways to arrange these days is given by the multinomial coefficient formula:
```
Number of arrangements = 7! / (3! × 2! × 2!)
```
Here:
- `7!` accounts for the total arrangements of all days.
- `3!`, `2!`, and `2!` account for the repeated days of types A, B, and C.
5. **Calculate the values**:
- `7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040`
- `3! = 3 × 2 × 1 = 6`
- `2! = 2 × 1 = 2` (for both type B and C)
Substituting into the formula:
```
Number of arrangements = 5040 / (6 × 2 × 2)
= 5040 / 24
= 210
```
6. **Answer**:
The total number of different timetables that can be compiled is **210**.
### Final Answer: **(b) 210**
