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There are 12 intermediate stations between two places A and B. The number of ways a train can be made to stop at 4 of these intermediate stations so that no two stopping stations are consecutive is:

(a) 792 ways
(b) 924 ways
(c) 1,056 ways
(d) 1,320 ways

User Shrewmouse
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1 Answer

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Final answer:

To solve this problem, we can use the concept of combinations. By choosing stations alternately and skipping consecutive stations, we can calculate the number of ways a train can stop at 4 intermediate stations between A and B. The correct answer is 924 ways.

Step-by-step explanation:

To solve this problem, we can use the concept of combinations. Since there are 12 intermediate stations, we need to choose 4 stations to stop at. However, we must ensure that no two stopping stations are consecutive.

Let's consider the 12 intermediate stations as slots to fill. We can start by selecting a station to be the first stopping station. We have 12 choices for this. Then, we need to choose 3 more stopping stations from the remaining 10 stations. To ensure no two stopping stations are consecutive, we alternate choosing a station and skipping the next one.

We have 10 choices for the second stopping station, 9 choices for the third stopping station, and 8 choices for the fourth stopping station. Therefore, the total number of ways to choose the stopping stations is 12 × 10 × 9 × 8 = 8,640.

However, since the order in which we choose the stopping stations doesn't matter, we need to divide this number by the number of ways to arrange the 4 chosen stations. The number of ways to arrange 4 stations is 4 factorial (4!). Therefore, the number of ways a train can be made to stop at 4 of these intermediate stations so that no two stopping stations are consecutive is:

8,640 ÷ 4! = 924 ways

Therefore answer is (b) 924 ways.

User NickAldwin
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