A train having 12 stations enroute has to be stoped at 4 stations. The number of ways it can be stopped if no two stoppings stations are consecutive

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Consider 4 identical things (AAAA) and 8 other identical things (BB….B) to be arranged in a row so that no two A’s are together.

$_B_B_B_B_B_B_B_B_$

There are 9 gaps between the B’s. The number of ways of doing this is $^{9} C_{4}$ . This is the final answer. For each such arrangement, we have one way of stopping the train at 4 stations (positions of A’s).

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