There are $3$ americans, $2$ britishers, $1$ portugese, $1$ chinese and they are allowed to sit around a circular table so that no two people of same nationality sit side by side. Answer is $3148$. I ve tried it many a times but...answer given in book is 3148
2026-04-04 15:13:05.1775315585
Combinatorics circular sitting problem
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The Americans must be placed: A _ _ A _ A _
There are $3!$ ways to do that.
There are four remaining people to put in the four seats. This can be done in $4!$ ways, two of which have the British in the two adjacent slots; these cases must be eliminated.
Hence $4! - 2 = 22$ cases where we do not care where the first person is seated (i.e., we can rotate the table). If we do care about orientation, we multiply by 6.
Answer: $3! (4!-2) 6/2 = 396$.