## sh3lsh one year ago How many distinct ways can we arrange four A’s and two B’s around a circle? Explanation preferred.

1. perl

You can do this by trial and error , or we can look up "circular permutations with repetitions allowed".

2. sh3lsh

Is there a concrete formula for this? I've found this site which referred to circular permutations by just $(n-1)!$ http://www.ditutor.com/combinatorics/circular_permutations.html

3. perl

that is for n distinct objects , here we have reptition

4. perl

non distinct objects

5. sh3lsh

I understand. I'm trying to find other material online that will help with this question. I'm afraid I don't know what to do to account for the repeated elements.

6. perl

I don't see a simple formula online either :) We can do it out , list all the possibilities

7. sh3lsh

http://artofproblemsolving.com/community/c6h521144 Oh, should I just attempt these problems through trial and error? Doing so results in 3.

8. sh3lsh

Do you understand why they get the 2 in 5 choose 2?

9. perl

Because there are 2 arrangements that are different up to rotation.

10. perl

do you understand how they got this aabbb = abbba = bbbaa = bbaab ababb = abbab

11. sh3lsh

It's in a circle, so you can rotate the terms a bit to make it equivalent. Correct?

12. perl

right

13. perl

Lets find the number of ways to arrange AAAABB , we know this is $$\Large \frac {6! } {4!~2! }$$ then we can delete those that are the same up to rotation

14. sh3lsh

How do we find the duplicates without enumeration?

15. perl

you would have to use that formula here scroll down to the last problem on this page http://www.artofproblemsolving.com/community/c6h539781

16. sh3lsh

Oh gosh, this is too much. I'll just have to enumerate.

17. perl

they did all the enumerations there . they got 5

18. perl

Let B = '1' and A = '2' $$112222 \sim 221122 \sim 222211 \\211222 \sim 222112 \sim 122221 \\121222 \sim 221212 \sim 122212 \\ 212122 \sim 222121 \sim 212221 \\122122 \sim 221221 \sim 212212$$

19. sh3lsh

Doesn't 212122 ~ 122212?

20. sh3lsh

Alrighty. I'm sort of understanding. I have to go through this example a little more. Thanks for the steps and help!

21. perl

a1 = 4 a2 = 2 define gcd (2,4) = d , which is 2 Let the circular arrangements be similiar if they only differ in rotation by N/d, here 6/2= 3. So the circular arrangements are similar if they different in rotation by 3 112222 and 221122 differ by 3 rotations , so they are similar. The point of making this equivalence class is then we can use that formula below.

22. perl

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