saeedt wrote:
Correct, How about when we want 2 people to sit together at any rate?
The answer may not among the choices
When you have, say, 5 people, A, B, C, D and E, and you want to know how many seating arrangements are possible in a row of 5 seats, the answer is 5!; you have 5 choices for the first seat, 4 for the second, and so on. Now, if A and B must sit next to each other, the answer will be smaller. First, if A is to the left of B, then we know AB must appear together in our seating arrangement, so we can think of AB as one person; there are 4! possible arrangements. Since B might be to the left of A, there are a further 4! arrangements, and the answer would be 4! + 4! = 2*4!.
Now in the case of a circular permutation, you can use the same reasoning. Suppose you have 5 people, and 5 seats around a circular table, and you want to know how many arrangements are possible when all that matters is how the people are seated relative to each other. With no restrictions, then it makes no difference where we seat person A since we only care about the order of the people relative to each other; we then have 4 choices for who is clockwise to A's left, 3 choices for the next seat, and so on, and the answer is 4!.
Finally, to get to your question, if, say, A and B need to be next to each other, and we have 5 seats around a circle, we have two possibilities. If B is clockwise to A's left, we can think of AB as one letter. We then have 3 choices for who is clockwise next to AB, 2 choices for the next seat, and 1 choice for the final seat, for 3! arrangements. If A is clockwise to B's left, we again have 3! arrangements, for a total of 2*3! arrangements.
Finally, I'd add that circular permutation questions are rare on the GMAT (though I have seen one or two), and they need to be very precisely worded. The question in the first post above does not make clear that it is asking about circular permutations. A permutation is circular *only* when we don't care which seat each person is in - we only care who is sitting next to each other. If you have a circular table, but one seat is next to the window, and you care who gets the window seat, then the permutation is *not* circular; it's just a standard seating arrangement question in that case. A properly worded circular permutation question will always include a phrase like 'relative to each other' to make clear that the permutation is indeed a circular one.
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