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(C) The total number of ways to choose $5$ stations out of $15$ is given by ${ }^{15} C_5$. To find the number of ways where no two stations are conse

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${ }^{15} C_5 - { }^{11} C_5$

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(C) The total number of ways to choose $5$ stations out of $15$ is given by ${ }^{15} C_5$. To find the number of ways where no two stations are consecutive,we use the gap method. Let the $10$ stations where the train does not stop be represented by $X$. $X_1 X_2 X_3 X_4 X_5 X_6 X_7 X_8 X_9 X_{10}$ There are $11$ possible gaps (including the ends) where we can place the $5$ stations where the train stops. The number of ways to choose $5$ gaps out of $11$ is ${ }^{11} C_5$. Thus,the number of ways in which the train stops at at least two consecutive stations is the total ways minus the ways where no two stations are consecutive: $= { }^{15} C_5 - { }^{11} C_5$.

There are $15$ stations on a train route and the train has to be stopped at exactly $5$ stations among these $15$ stations. If it stops at at least two consecutive stations,then the number of ways in which the train can be stopped is

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