how many ways can 7 children be arranged in a circular table

Question

how many ways can 7 children be arranged in a circular table where 2 chidren must be apart.

prep_gmat · Answer

(# of ways 7 children can be arranged around a circular table) - (# of ways the 2 children can be seated together)

# of ways 7 can be arranged in a circular pattern = 6!
# of ways 2 can be seated together = 2.5!

# of ways 2 children cannot be seated together in a circular pattern = 6! - 2.5!.

christoph · Answer

do you multiply by 2 because it is a permutation problem ? otherwise pls explain ! thx

HongHu · Answer

There are two ways to arrange the two boys who sit together, one way is for A to sit left to B, the other is for A to sit right to B.

Antmavel · Answer

Is this the OA ?
Please explain how did you get 2.5!

Thanks

christoph · Answer

take for example a and b as the two children. they sit in the first and second position. next to them there are 5*4*3*1 possible ways to sit the other children. we multiply by 2 because there are even more ways when a and b switch their seats.

Antmavel · Answer

sorry christoph, I am still lost on this one

I can't figure out the reasonning process

first, can you confirm if there are : 5*4*3*1 possible ways to sit the other children (like you've written) ? or 5*4*3*2*1 possible ways to sit the other children ? I am not sure I could answer right if I see this kind of questions again at GMAT test :oops:

christoph · Answer

oh a typo ! yes 5*4*3*2*1

first scenario: a b 5*4*3*2*1 = 5!
second scenario: b a 5*4*3*2*1 = 5!
=> 2*5!

Antmavel · Answer

i got it i read 2.5! (like 2,5) whereas your solution was 2*5! (2 multiplied by 5!) :roll:

sorry, now i totally understand it. Anyway, thank you for your time and for your explanation christoph, it was nice.

Arsene_Wenger · Answer

I'M completely lost here.

Let's say we have seats A-G. i.e. A,B,C,D,E,F,G
then, the no. of ways 2 people can stay together is:

AB, BC, CD, DE, EF, FG, GA (7.2 = 14 ways!)

christoph · Answer

this is a problem of circular permutation. imagine a circle with 3 seats A B C:

first arrangement is A B C
second is A C B

that is because the circle can be rotated. that is why we use (n-1)! instead of n! for the total ways of arrangements.

Vijo · Answer

I completely agree with prep_gmat's soln.

it will be !6 - (2*!5)

swath20 · Answer

One more for 6!- 5!2

Folaa3 · Answer

Please explain 2 * 5!
I don't get it.

MA · Answer

SUPPOSE THE TWO CHILDREN ARE X AND Y AND OTHERS ARE A,B,C,D, AND E.

The following positin is the way two children seat togather:
x-a-b-c-d-e-f-y or y-a-b-c-d-e-f-x.

therefore, the number of ways the children can be arranged: 2(5!).

=6!-2(5!)=480

how many ways can 7 children be arranged in a circular table

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