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## Homework Statement

You are given n identical As and n identical Bs. You are supposed to arrange them in a circle. How many unique arrangements are there?

## Homework Equations

Use of combinatorics

## The Attempt at a Solution

At first, I tried working out the answer if I arrange them in a row. The answer is (2*n)!/(n!*n!). Then I tried to multiply something to this to get the answer. However, I cannot find a way to do so. For example, I tried dividing it by 2*n, but this does not work, because not all the arrangements repeated 2*n times.

I worked out some answers below in case someone needs them.

for n = 1, there is only 1 way {AB}

for n = 2, there are 2 ways {AABB, ABAB}

for n = 3, there are 4 ways. {ABABAB, AABBAB, AABABB, AAABBB}

for n = 4, there are 9 ways if I did not count wrongly {AAAABBBB, AAABBBAB, AAABBABB, AAABABBB, AABAABBB, AABBAABB, AABABABB, AABABBAB, AABBABAB}

Sorry, my combinatorics is not that good. This is all I can do :(