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There are p intermediate stations on a r...

There are p intermediate stations on a railway line from one terminus to another . In how many ways a train can stop at 3 of these intermediate stations if no two of those stopping stations are to be consecutive ?

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Let x, y, z and t be the numbers of stations before first stoppage, between first and second stoppage , between second and third stoppages , after third stoppage respevtively , then
`x ge 0, y ge 1, z ge 1 , t ge 0 and x +y+z +t =p-3`.
Let `y=1+y_(1), and z=1+z_(1)`, so that `y_(1) ge 0 , z_(1) ge 0 and x+y_(1)+z_(1) + t =p-5.` The no. of non-negative integral solution of `x+y_(1)+z_(1)+t=p-5` is
`=""^(p-5+4-1)C_(4-1)=""^(p-2)C_(3)=((p-2)(p-3)(p-4))/(6)`
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