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Question
Between two junction stations A and B, there are 12 intermediate stations. The number of ways in which a train can be made to stop at 4 of these stations so that no two of these halting stations are consecutive, is?
Between two junction stations A and B, there are 12 intermediate stations. The number of ways in which a train can be made to stop at 4 of these stations so that no two of these halting stations are consecutive, is?
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Solution
The correct option is B 9C4
Let x1 be the number of stations before the first halting station, x2 between first and second and x3 between second
and third, x4 between third and fourth and x5 on the right of 4th stations.
Then x1≥0,x5≥0,x2,x3,x4≥1 satisfying x1+x2+x3+x4+x5=8...(1)
The total number of ways is the number of solution of the above equation
Let y2=x2−1,y3=x3−1,y4=x4−1.
Then(1) reduces to x1+y2+y3+y4+x5=5,
Where y2,y3,y4≥0.
The number of solution of this equation is 5+5−1C5−1=9C4.
9C4
Let x1 be the number of stations before the first halting station, x2 between first and second and x3 between second
and third, x4 between third and fourth and x5 on the right of 4th stations.
Then x1≥0,x5≥0,x2,x3,x4≥1 satisfying x1+x2+x3+x4+x5=8...(1)
The total number of ways is the number of solution of the above equation
Let y2=x2−1,y3=x3−1,y4=x4−1.
Then(1) reduces to x1+y2+y3+y4+x5=5,
Where y2,y3,y4≥0.
The number of solution of this equation is 5+5−1C5−1=9C4.
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