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FInd the number of ways of selecting 3 stations from 8 such that no two stations are consecutive.

deepak
84 Points
4 years ago
let us consider the stations as no’s such as 1 for the 1st station and so on.

then the stations in order are

$1 2 3 4 5 678$

there exists tw cases when the stations one and eight are selected when one and eight are not selected cos station one and eight each have only one station near them

case 1
when 1 and 8 are not selected
then the no of ways in which 1st station  can be selected is 6
and the no of ways in which 2nd station can be selected is 3
and the no of ways in which 3rd station  can be selected is 0
as for every single sation picked two stations near it cannot be picked and the station itself cannot be picked again
therefore the no of combinations without 1 and 8 are zero

case two
when 1 is seleced as one of the stations and 8 is not selected
let the first sation to be selected be 1
then
the no of ways in which the 1st station can be selected is 1
the no of was in whcih the second station can be selected is 5 (as 2 , 8, 1 cannot be selected)
the no of ways in which the 3rd station can be selected is 2
therefor the total no of combinations here is 5x2x6=60 we multiply by six cos the three nos themselves can be mixed in 3! ways

case 3
when 8 is seleced as one of the stations and 1 is not selected
let the first sation to be selected be 8
then
the no of ways in which the 1st station can be selected is 1
the no of was in whcih the second station can be selected is 5 (as 7 , 8, 1 cannot be selected)
the no of ways in which the 3rd station can be selected is 2
therefor the total no of combinations here is 5x2x6=60 we multiply by six cos the three nos themselves can be mixed in 3! ways

case 4
when both 1 and 8 are selected
let the first station be 1 and the second station selected be 8
the no of ways in which the 1st station can be selected is 1
the no of was in whcih the second station can be selected is 1
the no of ways in which the 3rd station can be selected is 4 (as 1,8,7,2 cannot be selected)
therefor the total no of combinations here is 4x6 = 24 we multiply by six cos the three nos themselves can be mixed in 3! ways

therefore the total no of ways in which the event can take place is 60+60+24=144

hope this was uselful B|