Linear Track
There are 12 Stations in a row as shown in the figure. A train stops at 4 stations such that no stops are at consecutive stations.How many such selections are possible?
The answer is 126.
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1 solution
Moderator note:
For clarity, you should avoid reusing notation. The stations were already numbered with 1 to 12, and using "non-stopping stations are marked as 1, 2, ... 8" might give others the wrong impression.
Initially, let's remove the 4 stopping stations
Then we are left with 8 non-stopping stations (=12-4) as shown below
. 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 .
(non-stopping stations are marked as 1,2 ... 8)
Now there are 9 positions (as marked by . ) to place the 4 stopping stations
such that no two stopping stations are consecutive
This can be done in 9C4 ways
Hence, required number of ways = 9C4 =126