2019 AIME II Problems/Problem 5

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Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ people to be seated at the table under these conditions. Find the remainder when $N$ is divided by $1000$.


There are $4$ ambassadors and there are $6$ seats for them. So we consider the position of the blank seats. There are $15$ kinds of versions: If the two seats are adjacent to each other, there are $6$ options, and the ambassadors are sitting in four adjacent seats, and there are five seats that their advisors can sit. Choose any of them and the advisors’ seats are fixed, so there are $5$ kinds of solutions for the advisors to sit. And that’s a $6\cdot5$ if we don’t consider the order of the ambassadors. We can also get that if the blank seats are opposite, it will be $3\cdot9$, if they are not adjacent and not opposite, it will be $6\cdot8$. So the total is $24\cdot(6\cdot5+6\cdot8+3\cdot9)=2520$ And the remainder is $\boxed{520}$

See Also

2019 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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