Math Help - Factorials 2

1. Factorials 2

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?

this is my work, but does not lead me to the correct answer:

7! x 2 x 2 = 20160

The answer they have is 201600

2. Originally Posted by skeske1234
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 answer they have is 201600
That is not the answer that I get.
I done it three different ways and get 211680.

3. Originally Posted by Plato
That is not the answer that I get.
I done it three different ways and get 211680.

How did you get your answer of 211680

I have:

9! - (8! x 2) - (8! x 2)

4. Originally Posted by skeske1234
How did you get your answer of 211680
I have:
9! - (8! x 2) - (8! x 2)
If you do that way then you have removed the case where Henry sits between Wilson and Nancy, NHW or WHN, twice.
Using inclusion/exclusion we get:
$9!-4\cdot(8!)+2\cdot(7!)$