1. ## Permutations cont.

Six couples are to be seated around a circular table, in 12 chairs. How many ways can this be done if each couple is together?

2. ## Re: Permutations cont.

Originally Posted by azollner95
Six couples are to be seated around a circular table, in 12 chairs. How many ways can this be done if each couple is together?
Circular arrangements have a different count. There are $(N-1)!$ ways to arrange $N$ distinct items in a circle.
In this problem the six couples sitting together can be though of as six distinct items. But each couple can be seated in two ways(husband to the wife's right or visa-versa.) So what is the total?

3. ## Re: Permutations cont.

Would it simply be (12!) since you can continue to rotate them around?

4. ## Re: Permutations cont.

Originally Posted by azollner95
Would it simply be (12!) since you can continue to rotate them around?
usually when the problem specifies the arrangement is circular it means that rotations are indistinguishable from one another.

5. ## Re: Permutations cont.

Originally Posted by azollner95
Would it simply be (12!) since you can continue to rotate them around?
Absolutely NOT!

In this question we are arranging six objects(couples together) in a circular arrangement each object can be arranged in two ways.

ANSWER: $(5!)(2^6)$.

6. ## Re: Permutations cont.

I now understand where the 5! is coming from but why wouldn't you times (5!) by 2 instead of 2^6?

7. ## Re: Permutations cont.

Originally Posted by azollner95
I now understand where the 5! is coming from but why wouldn't you times (5!) by 2 instead of 2^6?
Because there two(2) ways to seat each couple: husband to the right of wife or wife to the right of husband. Six couples so $2^6$ ways.