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There are 20 intermediate stops on a route of a transport corporation bus. The number of ways in which the bus can stop at 6 of these intermediate stops such that no 2 stops are consecutive is ?

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Number of stopping stations = 6

Number of non-stopping stations = 14

If we lay down the 14 non-stopping stations then we will have 15 choices where the stopping stations can be and none of them would be consecutive.

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

* | 9 | * | 10 | * | 11 | * | 12 | * | 13 | * | 14 | * |

Here * are free positions and numbers are non-stopping stations.

So we have to select 6 places from 15,

i.e., $${15\choose6} = \frac{15\times14\times13\times12\times11\times10}{6\times5\times4\times3\times2\times1} = 5005$$

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