Seating Arrangement

Because it is asking for arrangements. nPr is used when order matters, nCr is used when order does not matter.

How does order matter here?

they could sit ABCDEF BADCEF ..etc

MBA.Aspirant wrote:How does order matter here?

they could sit ABCDEF BADCEF ..etc

The question says, "what's the number of possible arrangements", so I'd say it is a permutation. I admit that probability is not my strong point though, so perhaps someone who is stronger in the subject can weigh in.

the relative order in which people are seated matters, but the start and end point do not.

because for each permutation there will be exactly 5 duplicates (or 6 total), i think you just need to do 6! and divide it by 6 to get 120.

in this form of question, the only correct answer is
- if the start point matters it is 6!, otherwise 6!/6

Q: 6 people to be arranged around a circular table, what's the number of possible arrangements?
Why is this a ‘permutation’?

This would be a combination if order of arrangement did not matter (Example: Number of ways to select 4 out of 6).

Here the order of arrangement matters and hence it is a permutation.
6 people around a circular table
Permutation of 6 people = 6P6 = 6*5*4*3*2*1 = 720
***The arrangement differs because abcdef is different from acbdef, and so on…

BUT they are seated around a circular table => abcdef = bcdefa = cdefab = defabc = efabcd = fabcde ……...because there is no “starting point” for a circular arrangement!
=> For every 6 possible permutations only one is unique because of the circular positioning
=> Number of different possible seating arrangements = 720/6 = 120

You can generalize this principle for all simple circular arrangements of N objects to (N-1)P(N-1) permutations.
=> For an arrangement of 7 persons around a circular table, we can have 6P6 = 720 unique possibilities.

I hope this explanation helps!

Knewtonian wrote:Q: 6 people to be arranged around a circular table, what's the number of possible arrangements?
Why is this a ‘permutation’?

This would be a combination if order of arrangement did not matter (Example: Number of ways to select 4 out of 6).

Here the order of arrangement matters and hence it is a permutation.
6 people around a circular table
Permutation of 6 people = 6P6 = 6*5*4*3*2*1 = 720
***The arrangement differs because abcdef is different from acbdef, and so on…

BUT they are seated around a circular table => abcdef = bcdefa = cdefab = defabc = efabcd = fabcde ……...because there is no “starting point” for a circular arrangement!
=> For every 6 possible permutations only one is unique because of the circular positioning
=> Number of different possible seating arrangements = 720/6 = 120

You can generalize this principle for all simple circular arrangements of N objects to (N-1)P(N-1) permutations.
=> For an arrangement of 7 persons around a circular table, we can have 6P6 = 720 unique possibilities.

I hope this explanation helps!

this blew my mind, as I kept getting 720 . is this one of the trickier quant problems one would see on the GMAT? im just starting to study the math side now..

MBA.Aspirant wrote:6 people to be arranged around a circular table, what's the number of possible arrangements?

Let's say that the 6 people are ABCDEF.

If we were to count the ways to arrange ABCDEF in a line, the following would qualify as different arrangements:

ABCDEF
BCDEFA
CDEFAB
DEFABC
EFABCD
FABCDE

But when put around a table, all of the above would qualify as only one arrangement, because the clockwise order would always be the same: A-B-C-D-E-F. In all of the above, B would be directly to the right of A; C would be directly to the right of B; D would be directly to the right of C; E would be directly to the right of D; and F would be directly to the right of E.

Thus, the number of ways to arrange N people around a circular table is smaller than the number of ways to arrange the N people in a line:

Number of ways to arrange N people around a circular table = (N-1)!.

So given 6 people, there are (6-1) = 5! = 120 ways to arrange them around a circular table.

120 is the correct answer.
As this is circular arrangement ans is n!/n