There are 10 stations on a circular path. A train has to stop at 4 stations such that no two stations are adjacent. The number of such selections must be:

Options

A. 25
B. 35
C. 210
D. 50

Answer

To solve the problem of selecting 4 stations from 10 on a circular path such that no two stations are adjacent, we can follow these steps:

### Step 1: Understand the Circular Arrangement
Since the stations are arranged in a circle, we need to account for the circular nature of the arrangement. This means that if we choose one station, we cannot choose its adjacent stations.

### Step 2: Transform the Circular Problem to a Linear Problem
To simplify the problem, we can "break" the circle at one point. By fixing one station (let's say station 1), we can treat the remaining stations as a linear arrangement. This gives us 9 remaining stations (2 to 10).

### Step 3: Account for Non-Adjacent Selection
When we select 4 stations, we need to ensure that no two selected stations are adjacent. If we select 4 stations, we will have 4 gaps between them (since we cannot select adjacent stations). Additionally, we will have to account for the gaps created by the circular arrangement.

### Step 4: Calculate the Effective Stations
If we select 4 stations, we will have 4 gaps that must be filled with at least one unselected station to ensure that no two selected stations are adjacent. Thus, we can think of the problem as needing to place 4 selected stations and at least 4 unselected stations (one in each gap) among the remaining stations.

This means we have:
- Total stations = 10
- Selected stations = 4
- Minimum unselected stations needed = 4 (to separate the selected ones)

This leaves us with:
\[ 10 - 4 - 4 = 2 \]
unselected stations that can be placed freely.

### Step 5: Use Stars and Bars Method
Now, we need to distribute these 2 remaining unselected stations into the 5 gaps (1 before the first selected station, 3 between the selected stations, and 1 after the last selected station).

The number of ways to distribute \( k \) indistinguishable objects (unselected stations) into \( n \) distinguishable boxes (gaps) is given by the formula:
\[ \binom{n+k-1}{k} \]

In our case:
- \( n = 5 \) (the gaps)
- \( k = 2 \) (the remaining unselected stations)

Thus, we need to calculate:
\[ \binom{5+2-1}{2} = \binom{6}{2} \]

### Step 6: Calculate the Binomial Coefficient
Now we calculate \( \binom{6}{2} \):
\[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

### Conclusion
Thus, the number of ways to select 4 stations from 10 on a circular path such that no two stations are adjacent is **15**.

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