Question

A and B are two junctions with 31 stations between them. In how many ways can a train go from A to B while halting at 6 stations between them such that any consecutive pair of halts (including A and B) are separated by an odd number of stations?

Hint:

In problems involving selection with specific distance conditions, break the range into even and odd intervals, then use combinations to find the total number of selections.

Answer 1

Step 1: Understanding the problem.
We are given 31 stations between A and B. The train halts at 6 stations. So, the total number of stops is 8, including A and B. Step 2: Condition for odd separation.
The condition is that any two consecutive halts (including A and B) should be separated by an odd number of stations. In other words, the distance between two consecutive stops must be odd. Since we are halting at 6 stations, this condition divides the 31 stations into odd and even intervals. Step 3: Breaking the stations into even and odd groups.
For each halt, the station number can either be odd or even. The number of ways to select 6 stations from the 31 stations while ensuring that any two consecutive stops are separated by an odd number of stations can be determined using combinations. This leads to the number of possible arrangements being \( 600 \). Step 4: Conclusion.
Thus, the correct answer is 600.