A family consisting of one mother, one father, two daughters and a son is taking a road trip in a sedan. The sedan has two front seats and three back seats. If one of the parents must drive and the two daughters refuse to sit next to each other, how many possible seating arrangements are there?
28
32
48
60
120
I tried using slots method here.
The driver seat can be filled in 2 ways
The passenger seat can be filled in 4 ways.
The 3 rear seats can be filled in 3*2*1 ways.
All the different ways to fill the seats are : 2*4*3*2*1 = 48
Now, we have to remove all the ways in which two daughters, D1 and D2 sit together. I used the glue method as mentioned in the strategy guide - so, out of 4 possible seats (1 passenger and 3 rear), we have 3 people (considering both daughters as one unit) - of which 3! ways (6) of arranging are possible. Add another 6 to it to consider the possibility of D2 sitting next to D1 - which results in a total of 12 to be removed from the total number of possibilities (48)
48 - 12 = 36.
This is where I am stuck. I think there are a few other combinations to remove from this and I am not sure what they are. Could anyone please let me know if this is the right approach and if so, what the next step is. Any better approaches are definitely welcome.