Counting Question Concerning Circular Arrangements

[UltimateSomni]

Homework Statement

I have two questions. I'm not sure if I'm allowed to post two at once so I'll start with one

"Twenty boys and twenty girls are to take a ride on a Ferris wheel with twenty pods. How many ways can they be arranged if each pod is to contain one boy and boy girl"

Homework Equations

For circular arrangements, (n-1)! possible arrangements

The Attempt at a Solution

1. Fix the first boy, sort of the other 19 around him... 19! ways
2. Now, with all the spots set in terms of the first boy, sort the 20 girls... 20! ways
3. Thus, there are 20!*19! ways to sort them

I am pretty sure this is correct but I can't find my notes so I do not know for sure.

 

[andrewkirk]

Yes, that looks correct to me.

 

[UltimateSomni]

Yes, that looks correct to me.

All right, I thought so. Well we did a similar one that my notes give an odd answer for.

Homework Statement

"How many ways are there to seat 5 boys and a 5 girls at a round table so that boys and girls alternate?

Homework Equations

For circular arrangements, (n-1)! possible arrangements

The Attempt at a Solution

1. Fix the first boy, arrange the other 4 around him... 4!
2. With the spots set, arrange the 5 girls... 5!
3. In total, there are 5!*4! ways

But my notes say (5!)^2. Which one is right?

 

[andrewkirk]

Either one can be correct, depending on what we mean when we say that two seating plans are different. $(5!)^2$ is correct if making them all stand up and move one place to their left is regarded as changing the seating plan. If it isn't then $5!4!$ is the correct answer. In the Ferris wheel case, it was not regarded as a change, so $20!19!$ was the answer.