In how many ways can 6 people be seated around a circular table if

Question

In how many ways can 6 people be seated around a circular table if 2 particular people are always separated? (2 seating arrangements are considered different only when the positions of the people are different relative to each other.)

A. 24
B. 48
C. 72
D. 96
E. 120

Show Answer

Official Answer

C

A. 24
B. 48
C. 72
D. 96
E. 120

CrackverbalGMAT · Accepted Answer

If n people are seated in a circle on n similar chairs, then the total number of arrangements = (n - 1)!

Number of ways when they are always separated = Total seating possibilities - # of ways when they are always together.

Total possible ways = (6 - 1)! = 5! = 120

# of ways when they are always together: Assume these 2 people to be 1 unit. Then we have to arrange 5 units in a circle in (5 - 1)! = 4! = 24 ways.

Also these 2 people can interchange seats among themselves in 2! = 2 ways

Therefore total number of ways when they are always separate = 120 - (24 * 2) = 120 - 48 = 72

Option C

Arun Kumar

AlexTheTrainer · Answer

Two options

Option 1: use logic and limited knowledge to eliminate three answer choices in under a minute

Just need to know the formula for N circular arrangements = (N - 1)! = 5! = 120

This tells you answer choice E must be too big.

The question asks about only 2 out of 6 elements, which isn’t particularly restrictive. Logic dictates that the answer will be at least half the total number of arrangements.

Answers A and B are clearly too small.

Only C and D remain. 
..........
Option 2: the math

(Total number of arrangements of all 6) - (Number of arrangements where X & Y are paired)

Number of unique circular arrangements for N elements = (N - 1)! = 5! = 120

NOTE: (N - 1)! = N!/N

Corollary: Number of unique circular arrangements for N elements where specific elements are restricted = (Number of restricted arrangements)/N

.........

Number of unique arrangements for N elements where S specific elements are restricted = 
(# of arrangements for S) x (N - S)!

Number of unique circular arrangements for N elements where X & Y are paired together =
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MathRevolution · Answer

Total ways to arrange 6 people in a circle = (6 - 1)! = 120

Total ways to arrange 6 people in a circle such that 2 people are always together = (5 - 1)! = 24 and these two people can be arranged in 2! ways. Therefore, total 24 * 2 = 48 ways

Total ways to arrange 6 people in a circle if 2 particular people are always separated: 120 - 48 = 72

Answer C

CrackverbalGMAT · Answer

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

tramobile · Answer

we have already taken out the relative position i.e AB & BA are out in 120, so it should not be subtracted
e.g. ABCDEF or BACDEF only one of them is counted in 120, but while subtracting you are using both case.
 CrackVerbal

CrackverbalGMAT · Answer

tramobile

You are considering the arrangement ABCDEF and BACDEF as a single arrangement and think that it is counted only once in the total 120 arrangements.

Let's analyze your above statement in detail.

Remember, 2 seating arrangements are considered different when the positions of the people are different relative to each other.

The example you have mentioned, i.e ABCDEF and BACDEF are 2 different arrangements because, in the arrangement ABCDEF, B is sitting between A and C in the circular table while in BACDEF, B will be sitting between F and A. Here, the position of B is different in both cases, Hence it should be counted as 2 different arrangements.

I hope this is clear for you and that's the reason we are subtracting both cases i.e 2!*4! from the total 120 arrangements.

Let's us know if you have any further queries.

Thanks,
Clifin Francis,
GMAT Mentor

Nalapatricia · Answer

In how many ways can 6 people be seated around a circular table if 2 particular people are always separated? (2 seating arrangements are considered different only when the positions of the people are different relative to each other.)
 
First, ways to arranged 6 people:
5! = 5*4*3*2*1 = 120

Ways where those 2 people are together (Considered as a one entity so) = 4! = 4*3*2*1 = 24
Now, consider that those person together could be P1 and P2 and you have 2 ways to sit them together, so we have to multiply 24*2 = 48

Now we have that total ways - ways that are restricted = 120 - 48 = 72

Therefore, correct answer:
C. 72

In how many ways can 6 people be seated around a circular table if

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In how many ways can 6 people be seated around a circular table if

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