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

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

## 1 Answers

6 years ago

let us consider the stations as no’s such as 1 for the 1

^{st}station and so on.then the stations in order are

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 1

^{st}station can be selected is 6and the no of ways in which 2

^{nd}station can be selected is 3and the no of ways in which 3

^{rd}station can be selected is 0as 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 1

^{st}station can be selected is 1the 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 3

^{rd}station can be selected is 2therefor 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 1

^{st}station can be selected is 1the 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 3

^{rd}station can be selected is 2therefor 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 1

^{st}station can be selected is 1the no of was in whcih the second station can be selected is 1

the no of ways in which the 3

^{rd}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|