gateacademy test series

Question

There are 12 stations on a rail route. How many ways a special train can stop at 4 of these stations, so that no two stops are consecutive stations? please explain in detail

Comments

My ans is 126

Is it correct?

Answer 1

Now as there are 12 Stations out which Train has to be stopped at 4 stations so…

Let   1 2 3 4 5 6 7 8 9 10 11 12 ..these are the stations ...Out  of them we have to take 4 stations such that they are non-adjacent now ..

At First count Non-Adjacent Places …Just Think..🤔…..

Suppose train stops at 4 non-adjacent  …... so left stations are 8..

1 2 3 4 5 6 7 8….Now these 8 stations would help us to make that 4 stations non-adjacent …?

Here Non-Adjacent Places are marked as *

*1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 *

Yahoo!😁..there are 9 stars hence 9 non-adjacent...Out these 9 we have to select 4 stations ...as on stopping we get total 12 stations back…

Now Here should we apply Permutation or Combination??😩...No Worries ..

Let  apply 9P4  ...but here 4 stations that are selected are in order...as Permutation takes first 4 non-adjacent places  as 24 ways similar with all so to count once we divide by 4! …

9P4/4!= 9!/(5!*4!)==126

Or choose directly 9C4=126