- #1
gaganpreetsingh
- 24
- 0
There are n stations between two cities X and Y. At train is to stop at three of these n stations. Find the probability that no two of these three stations are consecutive.
This is what I did:
Total number of possibilities nc3 .
Now suppose that the stations train stops at is such that exactly two of these are consecutive.
(1,2) then it may stop at (4,5,6…..n) and no. of possibilities are n-3
(2,3) then it may stop at (5,6,7…..n) and no. of possibilities are n-4
…………………………………….
(n-3,n-2) then it may stop only at n and no. of possibilities are 1
So the number of ways in which the train may stop at exactly two consecutive stations is
(n-3) + (n-4) + ……… + 1 n-3 terms
using formula of A.P. [n/2 (a+l)] n is no. of terms, a is first term, l last ]
possibilities are (n-3)(n-2)/2
Now if exactly 3 stations are consecutive then (1,2,3) (2,3,4)….. (n-2,n-2,n)
Hence here the no. of possibilities are n-2
So the things I have to exclude are (n-3)(n-2)/2 + (n-2)
Which is equal to (n-1)(n-2)/2
So we have Pr{event} = 1 – (n-1)(n-2)/2*nC3 = (n-3)/n
But I am not getting the right answer. Any help where my analysis is wrong?
This is what I did:
Total number of possibilities nc3 .
Now suppose that the stations train stops at is such that exactly two of these are consecutive.
(1,2) then it may stop at (4,5,6…..n) and no. of possibilities are n-3
(2,3) then it may stop at (5,6,7…..n) and no. of possibilities are n-4
…………………………………….
(n-3,n-2) then it may stop only at n and no. of possibilities are 1
So the number of ways in which the train may stop at exactly two consecutive stations is
(n-3) + (n-4) + ……… + 1 n-3 terms
using formula of A.P. [n/2 (a+l)] n is no. of terms, a is first term, l last ]
possibilities are (n-3)(n-2)/2
Now if exactly 3 stations are consecutive then (1,2,3) (2,3,4)….. (n-2,n-2,n)
Hence here the no. of possibilities are n-2
So the things I have to exclude are (n-3)(n-2)/2 + (n-2)
Which is equal to (n-1)(n-2)/2
So we have Pr{event} = 1 – (n-1)(n-2)/2*nC3 = (n-3)/n
But I am not getting the right answer. Any help where my analysis is wrong?