Between two junction stations, there are 12 sub stations. In how many ways a train can stop at 4 of these sub stations such that no two of these halting sub stations are consecutive?
I have no idea from where to start this problem....been thinking of it for quite some time....One possible way would be to subtract the no. of ways in the train halts at at least 2 consecutive stations then add back no. of ways it can stop at at least 3 consecutive stations and finally subtract the no. of ways it can stop at exactly four consecutive stations; from the total no. of ways it can stop at four stations out of the 12 given stations....
But apparently the answer I get from this is wrong...can anyone help me out?? Thanks in advance!!
Rephrased you are asked to find the number of sums: $$n_1+n_2+n_3+n_4+n_5=8$$ under the extra condition that $n_1,n_5$ are nonnegative integers and $n_2,n_3,n_4$ are positive integers.
Here e.g. $n_2$ stands for the number of substations in between the first and the second substation where the train stops.
That comes to the same as finding the number of sums: $$m_1+m_2+m_3+m_4+m_5=5$$ under the extra condition that $m_1,m_2,m_3,m_4,m_5$ are nonnegative integers.
This can be solved with stars and bars.
Give it a try.