Question

There are 6 points on a circle. Each point is unique. Five events occur at five of these points, one after the other and each one at a different point. In how many ways can they occur such that no two successive events occur at adjacent points?

Answer 1

Answer:

Let us denote the 6 points on the circle by digits 1,2,3,4,5,6. It is to be noted that 1 and 6 are adjacent points, as they are on the circle. Let us consider the first 2 non-adjacent points where events occur in succession. There are 18 such 2-digit sequences, where the first 2 events occur and they are 13,14,15,24,25,26,31,35,36,41,42,46,51,52,53,62,63,64.

Now consider the sequence 13, where the 1st and 2nd event occurs. We are left with 3 events to occur and they can occur at 3 points out of the remaining 4 points 2,4,5,6. The 3 points out of 4 can be chosen in 4C3 or 4 ways. With 13 as the initial sequence of occurrence, we have 4 sets of 3 points, namely, 245,246,256 and 456. We have to reorder each of these 4 sets and place them after 13, to represent the 5-digit sequence of events. For example 245 needs to be reordered to 524 to get the event sequence 13524, such that successive events do not occur at adjacent points. Following is the list of the 5-digit sequences of events, which are 84 in number

Step-by-step explanation:

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