my best guess is as follows:
driver's seat has 4 possibilities because there are 4 people and one of them has to be in the driver's seat.
that leaves 6 seats for 3 people.
looking at the first of the 6 seats, there are 3 choices there because there are 3 people left to choose that seat.
looking at the second of the 6 seats, there are 2 choices there because there are 2 people left to choose that seat.
looking at the third seat, there is 1 choice left there because there is one person looking for that seat.
so far you have 4 * 3! possibilities which is equal to 4 * 3 * 2 * 1 = 24.
now if you look at the 6 seats that aren't the driver's seat, only 3 out of the 6 have been chosen.
that, i believe, gets into how many sets of 3 can you make out of 6 where order is not important which is the combination formula.
that formula is 6C3 = (6*5*4) / (1*2*3) which results in 20 possible combinations.
your total number of possibilities is equal to 24 * 20 which would be equal to 480 possible arrangements.
this is extremely difficult to visualize.
i used much smaller examples to confirm the formula was reasonable.
i'll do one of them so you can see the logic i used.
assume 4 seats with 3 people.
number of possible choices for the driver's seat is 3.
that leaves 3 seats to be filled with the 2 remaining people.
the first seat has 2 choices and the second seat has 1 choice.
so far you have 3 * 2 * 1 = 6 possible choices.
since there are 3 seats and only 2 of them can be used at a time, the possible combinations of 2 seats at a time from 3 seats is 3C2 which is equal to 3.
your solution should be 6 * 3 = 18 possible ways the seats can be filled.
this is a big number but not that big that we can't confirm that the formula gave us the correct answer.
i'll do that below:
driver seat seat 1 seat 2 seat 3
people are a, b, c
x is the empty seat
1 a b c x
2 a c b x
3 a b x c
4 a c x b
5 a x b c
6 a x c b
this is the first set with a in the driver's seat.
the second set will have b in the driver's seat.
the third set will have c in the driver's seat.
the total should be 18.
i'll do the second set of 6 so you can see how it works.
7 b a c x
8 b c a x
9 b a x c
10 b c x a
11 b x a c
12 b x c a
another 6 possible combinations will result from placing c in the driver's seat.
the total is 18.
the formula checks out.
i used it with 5 seats and believe it worked there as well unless i made a mistake.
that's why i think it's good.
so we'll do your problem again just to make sure it was calculated correctly.
you have 7 seats and 4 people.
number of possible ways to fill the driver's seat is 4.
number of possible ways to fill the first 3 seats is 3! = 6.
only 3 of the 6 seats are filled.
number of possible ways to fill the 6 seats 3 at a time is 6C3 = 20.
total number of ways is 4 * 6 * 20 = 480.
without doing an exhaustive investigation i'm pretty sure the solutions is good.
if you agree, then that's the way to go.