Between Shirdee and Khandala, there are 10 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. 35
B. 70
C. 105
D. 10

Answer

To solve the problem of how many ways a train can stop at 4 out of 10 intermediate stations such that no two stops are consecutive, we can follow these steps:

### Step 1: Understanding the Problem
We have 10 intermediate stations and we want to choose 4 stations for the train to stop at, ensuring that no two chosen stations are consecutive.

### Step 2: Setting Up Variables
Let’s denote:
- \( x_1 \): Number of stations before the first stop.
- \( x_2 \): Number of stations between the first and second stop.
- \( x_3 \): Number of stations between the second and third stop.
- \( x_4 \): Number of stations between the third and fourth stop.
- \( x_5 \): Number of stations after the fourth stop.

### Step 3: Constraints
Since no two stops can be consecutive:
- \( x_2, x_3, x_4 \) must be at least 1 (since there must be at least one station between each stop).
- \( x_1 \) and \( x_5 \) can be 0 or more (there can be no restrictions before the first stop and after the last stop).

### Step 4: Adjusting for Constraints
To account for the requirement that \( x_2, x_3, x_4 \geq 1 \), we can redefine these variables:
- Let \( y_2 = x_2 - 1 \)
- Let \( y_3 = x_3 - 1 \)
- Let \( y_4 = x_4 - 1 \)

Now, \( y_2, y_3, y_4 \geq 0 \).

### Step 5: Formulating the Equation
The total number of stations can be expressed as:
\[ x_1 + (y_2 + 1) + (y_3 + 1) + (y_4 + 1) + x_5 = 10 \]
This simplifies to:
\[ x_1 + y_2 + y_3 + y_4 + x_5 = 7 \]

### Step 6: Counting Solutions
Now we need to find the number of non-negative integer solutions to the equation:
\[ x_1 + y_2 + y_3 + y_4 + x_5 = 7 \]
This is a classic "stars and bars" problem, where the number of solutions is given by:
\[ \text{Number of solutions} = \binom{n+k-1}{k-1} \]
where \( n \) is the total we want (7) and \( k \) is the number of variables (5).

Thus, we have:
\[ \text{Number of solutions} = \binom{7 + 5 - 1}{5 - 1} = \binom{11}{4} \]

### Step 7: Calculating the Binomial Coefficient
Calculating \( \binom{11}{4} \):
\[ \binom{11}{4} = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} = \frac{7920}{24} = 330 \]

### Final Answer
The number of ways in which the train can stop at 4 of these stations such that no two halting stations are consecutive is **330**.

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