I'm not very good at this, so take my explanation with a grain of salt. I'd also love to see other opinions!
Out of the ten spots, you get:
- three "pretentious" passengers
- seven mobile passengers
Let's tackle the first three:
a. the guy who can't travel backwards is fixed on the front-facing side, giving him 5 possible positions on that one
b. the two who won't travel facing the engine will have to sit on the other side. Since they're picky, they'll probably want to be seated first and will choose between the 5 chairs available: 5P2 = 20
The other seven passengers will sit in the remaining seven seats. They can do so in 7! = 5040 ways.
So I got 5*20*5040 = 504000 ways. Of course, I realize that this is an insanely large number, so, again, I may be wrong.
