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. `""^(9)C_(4)`
B. `""^(12)C_(4)-4`
C. `""^(9)C_(5)`
D. `(9)/(5)*""^(8)C_(4)`

Answer

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

### Step-by-Step Solution:

1. **Identify the Total Stations**:
We have 12 intermediate stations between junctions A and B.

2. **Determine the Number of Stops**:
The train is to stop at 4 of these stations.

3. **Calculate Non-Stopping Stations**:
Since the train stops at 4 stations, the number of non-stopping stations will be:
\[ \text{Non-stopping stations} = 12 - 4 = 8 \]

4. **Account for the Condition of Non-Consecutiveness**:
To ensure that no two halting stations are consecutive, we can visualize the arrangement. If we denote the halting stations as H and the non-halting stations as N, we can think of the non-halting stations as creating gaps where halting stations can be placed.

For example, if we have 8 non-stopping stations (N), they can be arranged as:
\[ N N N N N N N N \]
This arrangement creates 9 gaps (including the ends) where we can place the halting stations. The gaps are:
- Before the first N
- Between each pair of N's (7 gaps)
- After the last N

5. **Choose Gaps for Halting Stations**:
We need to choose 4 out of these 9 gaps to place our halting stations. The number of ways to choose 4 gaps from 9 is given by the combination formula:
\[ \binom{9}{4} \]

6. **Calculate the Combination**:
Now we compute \(\binom{9}{4}\):
\[ \binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126 \]

### Final Answer:
The number of ways in which the train can stop at 4 stations such that no two halting stations are consecutive is **126**.

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