A railcar compartment has 8 seats

[Guest]

A railcar compartment has 8 seats; 4 facing the engine of the train and 4 facing away from the engine. Of the right passengers, 3 prefer to face away from the engine, 2 prefer to face the engine, and the others have no preference. In how many ways can the passengers be seated if the order on each side of the car is disregarded?

I have done: 4C3 to account for the "face away engine ppl"
and 4C2 for the "face the engine ppl"

I know theres 3 people who dont care where they sit. And I dont know what to do next.

 

[Denis]

4! * 4! * 3 = 1728

Number the passengers 1 to 8; 4, 7 and 8 sit anywhere; then:

1, 2, 3, (4) : 4! ways to arrange; for each of those:
5, 6, (7), (8): 4! ways to arrange

4! * 4! = 576

Then (7) trades places with (4) : another 576
Then (8) trades placed with (7) : another 576

576 * 3 = 1728

 

[pka]

A railcar compartment has 8 seats; 4 facing the engine of the train and 4 facing away from the engine. Of the right passengers, 3 prefer to face away from the engine, 2 prefer to face the engine, and the others have no preference. In how many ways can the passengers be seated if the order on each side of the car is disregarded?

If indeed we disregard order but honor ‘facing’ preferences then there are only 3 ways to seat the passengers with no preference.