If we have 8 people in a meeting, in how many ways they can sit around

Question

If we have 8 people in a meeting, in how many ways they can sit around a table?

A) 10*9!
B) 9*8!
C) 8*7!
D) 7*6!
E) 6*5!

IanStewart · Accepted Answer

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.

fluke · Answer

If there are "n" people sitting around a table, there are "(n-1)!" possible arrangements:

Here, n=8
(n-1)!=7!=7*6!

Ans:"D"

subhashghosh · Answer

(8-1)! = 7! = 7*6!
Answer - D

saeedt · Answer

Correct, How about when we want 2 people to sit together at any rate?
The answer may not among the choices

saeedt · Answer

Great explanation IanStewart,

Yes, I thought the question can imply a circular permutation, so no need for an elaborate GMAT-like permutation. thanks for the explanation

Warlock007 · Answer

I think the answer would be (8-2)! x 2!(the ways to arrange two persons)=2 x 6!

ravsg · Answer

for original question, if it was not circular, then 2* 8!?

saeedt · Answer

Warlock007, Excellent, 2! (to people sitting together) * 6! ((8-6)!)= 2!*6!

I'm sorry the other two answers aren't correct

Warlock007 · Answer

Dont forget to give kudos Dude!!!!!!! :-D

gmat1220 · Answer

Hi Ian

I believe if AB are not restricted to sit together then in that case - the number of ways in which A is to the left of B is 5!/2. Similarly there are 5!/2 ways in which A is to the right of B. Can you pls confirm this?

Thanks

IanStewart · Answer

Yes that's right. If you are asked how many ways 5 people can line up so that A is  somewhere  to the left of B (not necessarily immediately next to B), there are 5! ways they could line up with no restrictions, and in half of those lineups A is to the left of B, and in half of those lineups, A is to the right of B, so there are 5!/2 lineups in which A is somewhere to the left of B.

Spidy001 · Answer

Answer is D.

7! = 7*6!

Lucky2783 · Answer

Answer (8-1)! or 7*6! Answer D

joelbr · Answer

Circular Permutation: n!/n

Therefore, 8!/8 = 7! (or 7*6!) (D)

Archit3110 · Answer

circular table n-1! ; n=8 
so arrangement possible 7! or say 7*6! 
IMO D

EgmatQuantExpert · Answer

Solution

Given  
 • 8 people in a meeting.

To find  
 • Number of ways in which 8 people can sit around a table.

Approach and Working out   
Sitting around a table is equivalent to circularly arranging the 8 people.
Thus, the number of arrangements = (8-1)! = 7! = 7*6!

Correct Answer: Option D

EgmatQuantExpert · Answer

Solution

Given  
 • 8 people in a meeting.

To find  
 • Number of ways in which 8 people can sit around a table.

Approach and Working out   
Sitting around a table is equivalent to circularly arranging the 8 people.
Thus, the number of arrangements = (8-1)! = 7! = 7*6!

Correct Answer: Option D

bumpbot · Answer

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If we have 8 people in a meeting, in how many ways they can sit around

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If we have 8 people in a meeting, in how many ways they can sit around

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