There are 8 railway stations along a railwag line.In how many ways can a train be stopped at 3 of these stations such that no two of them are consecutive.
Answers
Station 1, 2, 3, 4, 5, 6, 7 and 8.
The Train “Must” Stop at 3 Stations.
Answer : 20 combinations of 3 non consecutive stops.
So for just 20 combinations rather than use the combinations formulae I will explain the combinations below for how we arrive at this,
Station 1
Lets say the First Station Stop is Station 1.
Now you must choose second Station which cannot be Station 2.
And the Second Station cannot be Station 7 or 8. Because Choosing Station 7 means you can’t choose 8 as it’s consecutive. And you cannot choose Station 8 as you need to Choose the Third Station as well.
So we are left to choose Either Station 3, 4 , 5 or 6.
A choice of the Third Station becomes easier with any of these choices.
If we choose
Station 1 and then Station 3 , then we have 4 other choices for the THIRD Stop.
Station 1 and then Station 4 , then we have 3 other choices for the THIRD Stop.
Station 1 and then Station 5 , then we have 2 other choices for the THIRD Stop.
Station 1 and then Station 6 , then we have 1 other Choices for the THIRD Stop.
So in total IF we the FIRST Stop is STATION 1 , we can have — 4 + 3 +2 + 1= 10 Choices for Station Combinations.
Now you will say thats TOO MUCH Counting are we supposed to count from Station 1 to 8 for the Choice of the FIRST STOP.
No !
in fact we cannot go above STATION 4 for the CHOICE of the FIRST Stop because there are only 8 station, we have to Choose 3 and we cannot choose consecutive stations. Also we can only count forward not backwards
So if we choose STATION 2 as our First Stop
Then we have a choice of Station 4 , 5 or 6 as our SECOND Stop and easy to Choose the Third Stop.
Choice for 2nd Station as First STOP
2, 4, 6
2, 4, 7
2,4, 8.
Then choice of 5th station as the SECOND STOP
2, 5, 7,
2, 5, 8
And last of all 2, 6, 8.
Hence we have a total of 6 COMBINATION if the 2nd STATION is our FIRST STOP
3 Station as First STOP
3, 5, 7
3, 5, 8
3, 6, 8
Only 3 Choices above ! Not so much
And 4th Station as FIRST STOP
4, 6 ,8 - ONLY ONLY ONCE CHOICE
So totalling up all
10 + 6 + 3 + 1 = 20 COMBINATIONS.☺️☺️☺️