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Question

A train goes from A to B. There are 31 stations on the way. The train has to stop at any 6 stations but the number of stations between any 2 halts should be odd. In how many ways can the train halt?

Answer 1

Total number of ways train can stop=³¹C₆

Number of ways train can stop on station where difference between them is even=

(when train always stop on even number of station ) +(when train always stop on odd number of station)=¹⁶C₆ +¹⁶C₆=2*¹⁶C₆

So number of ways train stop on station whose difference is odd=³¹C₆  - 2*¹⁶C₆