There are 10 stations enroute. A train has to be stopped at 3 of them. Let N be the ways in which the train can be stopped if atleast two of the stopping stations are consecutive. Find the value of `sqrt(N)`.

Answer

To solve the problem, we need to find the number of ways a train can stop at 3 out of 10 stations, ensuring that at least two of the stopping stations are consecutive. We will break this down into two cases:

1. **Case 1**: All three stations are consecutive.
2. **Case 2**: Exactly two stations are consecutive, and the third is not.

### Step 1: Calculate the number of ways to stop at 3 consecutive stations.

If the train stops at 3 consecutive stations, we can represent the stations as follows:

- If the train stops at stations \(S_i, S_{i+1}, S_{i+2}\), where \(i\) can range from 1 to 8 (i.e., \(S_1, S_2, S_3\) to \(S_8, S_9, S_{10}\)), we have a total of 8 options.

Thus, the number of ways to stop at 3 consecutive stations is:
\[ \text{Ways for Case 1} = 8 \]

### Step 2: Calculate the number of ways to stop at exactly 2 consecutive stations.

For this case, we can denote the two consecutive stations as \(S_i, S_{i+1}\) and the third station as \(S_j\), where \(j\) is not consecutive to \(i\) or \(i+1\).

- The two consecutive stations can be chosen in 9 different pairs: (S1, S2), (S2, S3), ..., (S9, S10).
- For each pair of consecutive stations, we need to choose a third station that is not adjacent to them.

Let’s analyze the pairs:
- If we choose (S1, S2), the third station can be S4 to S10 (7 options).
- If we choose (S2, S3), the third station can be S5 to S10 (6 options).
- If we choose (S3, S4), the third station can be S6 to S10 (5 options).
- If we choose (S4, S5), the third station can be S6 to S10 (5 options).
- If we choose (S5, S6), the third station can be S1, S2, S8, S9, S10 (5 options).
- If we choose (S6, S7), the third station can be S1, S2, S3, S8, S9, S10 (5 options).
- If we choose (S7, S8), the third station can be S1, S2, S3, S4, S10 (5 options).
- If we choose (S8, S9), the third station can be S1, S2, S3, S4, S5 (5 options).
- If we choose (S9, S10), the third station can be S1 to S7 (7 options).

Now, we can sum these up:
- For (S1, S2): 7 options
- For (S2, S3): 6 options
- For (S3, S4): 5 options
- For (S4, S5): 5 options
- For (S5, S6): 5 options
- For (S6, S7): 5 options
- For (S7, S8): 5 options
- For (S8, S9): 5 options
- For (S9, S10): 7 options

Calculating the total:
\[ \text{Ways for Case 2} = 7 + 6 + 5 + 5 + 5 + 5 + 5 + 5 + 7 = 56 \]

### Step 3: Combine the results from both cases.

Now, we can find the total number of ways \(N\) in which the train can stop at 3 stations with at least two consecutive:
\[ N = \text{Ways for Case 1} + \text{Ways for Case 2} = 8 + 56 = 64 \]

### Step 4: Find the value of \(\sqrt{N}\).

Finally, we need to find \(\sqrt{N}\):
\[ \sqrt{N} = \sqrt{64} = 8 \]

Thus, the final answer is:
\[ \boxed{8} \]