The number of circular permutations of n objects = (n-1)!
The total number of ways in which 6 people can be arranged around a circle is therefore 5! = 120, which is not our answer since this is not what we want. Answer option E can be eliminated.
Since we want to find the number of arrangements where 2 people are always separated, we can find the total arrangements (i.e. 120) and then subtract the number of arrangements where these two people are together.
Number of permutations where A and B (say) are together = Total permutations – Number of permutations where A and B are together
Number of permutations where A and B are together:Since A and B have to be together, we consider them as ONE object. So, we now have a total of 5 objects (including the group of A&B). 5 objects can be arranged around a circle in 4! = 24 ways.
But, for each of these 24 ways, A and B can be arranged in 2! Ways.
Therefore, number of permutations where A and B are together = 4! * 2! = 48.
So, required number of permutations = 120 – 48 = 72.
The correct answer option is C.
In P&C questions with restrictions, using the backdoor approach saves time and effort and also ensures that we don’t get muddled in the various cases that the forward approach entails. Solving this question would be infinitely more difficult if we went about taking cases for organizing A and B separately.
Hope that helps!
Aravind B T
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Crackverbal Prep Team
www.crackverbal.com