# Ten people are seated at a rectangular table - Permutations homework

I got the following question for homework.

Ten people are to be seated at a rectangular table for dinner. Tanya will sit at the head of the table. Henry must not sit beside either Wilson or Nancy. In how many ways can the people be seated for dinner?

The approach I have used is to take $9!$ and subtract it from the seats where Henry, Wilson or Nancy can sit. I end up getting the answer $211680$ but for some reason, the back of the book says $201 600$. What am I doing incorrectly?

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This looks like an error in the book: your answer appears to be correct.

We can count the allowed seatings directly. Suppose that Henry sits next to Tanya. Then there is one seat forbidden to Wilson and Nancy, so there are $\binom72$ ways to choose seats for Wilson and Nancy. There are $2$ ways to seat Wilson and Nancy in those $2$ seats and $6!$ ways to seat the unnamed people. Finally, Henry can be on either side of Tanya, so there are altogether $2\cdot2\cdot6!\cdot\binom72$ acceptable seatings with Henry next to Tanya.

Now suppose that Henry is not seated next to Tanya. Then there are $2$ seats forbidden to Wilson and Nancy, so there are $\binom62$ ways to choose their seats. As before there are $2$ ways to seat them in those two seats and $6!$ ways to seat the unnamed people. Finally, there are $7$ possible choices for Henry’s seat, so there are altogether $7\cdot2\cdot6!\cdot\binom62$ acceptable seatings with Henry not next to Tanya. Altogether, then, there are

$$2\cdot6!\left(2\binom72+7\binom62\right)=1440(42+105)=211,680$$

acceptable seatings.

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Using inclusion-exclusion, it's possible to see where the error in the back of the book may have come from.

With Tanya assigned to the head of table, there are $9!$ ways to seat the other people with no restrictions. Among these, there $2\cdot8!$ ways in which Henry sits beside Wilson (the factor of $2$ corresponds to which one sits to the left of the other), and another $2\cdot8!$ ways in which Henry sits beside Nancy. Subtracting these, we get the book's answer

$$9!-2\cdot8!-2\cdot8!=201600$$

But this double-subtracts the $2\cdot7!$ ways in which Henry sits between Wilson and Nancy. So the correct answer is

$$9!-2\cdot8!-2\cdot8!+2\cdot7!=211680$$

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