Between two junction stations A and B, there are 12 intermediate stations. The number of ways in which a train can be made to stop at 4 of these stations so that no two of these halting stations are consecutive, is

Options

A. `""^(8)C_(4)`
B. `""^(9)C_(4)`
C. `""^(12)C_(4)-4`
D. none of these

Answer

To solve the problem of how many ways a train can stop at 4 out of 12 intermediate stations such that no two stopping stations are consecutive, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Total Stations and Stopping Stations**:
- There are a total of 12 intermediate stations.
- The train needs to stop at 4 of these stations.

2. **Calculate Remaining Stations**:
- If the train stops at 4 stations, the number of stations where the train will not stop is:
\[ \text{Remaining Stations} = 12 - 4 = 8 \]

3. **Understanding Non-Consecutive Stops**:
- To ensure that no two stopping stations are consecutive, we can visualize the 8 non-stopping stations as separators between the stopping stations.
- If we place the 8 non-stopping stations, they create gaps where the stopping stations can be placed.

4. **Count Gaps for Stopping Stations**:
- With 8 non-stopping stations, there are 9 possible gaps (including the ends) where the stopping stations can be placed:
- Gap before the first non-stopping station
- Gap between each pair of non-stopping stations (7 gaps)
- Gap after the last non-stopping station
- This gives us a total of:
\[ \text{Total Gaps} = 8 + 1 = 9 \]

5. **Choosing Gaps for Stopping Stations**:
- We need to choose 4 out of these 9 gaps to place the stopping stations. The number of ways to select 4 gaps from 9 is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
- Here, \( n = 9 \) and \( r = 4 \):
\[ \text{Ways} = \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]

6. **Final Answer**:
- Therefore, the number of ways in which the train can stop at 4 stations such that no two are consecutive is:
\[ \boxed{126} \]