Combinatorics using slots and anagrams

[psps]

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.

[michael_shaunn]

hi,
i think the solution is 32 and here's how i got it.
you are right that the total number of ways to make them sit is 48.

However,the typical part is to calculate the number of ways in which the sisters can be made to sit together.
To calculate that,here's how i proceeded,
1.The driver sit can be filled in 2 ways(either the mother or the father)
2.The daughters can be made to sit together only at the rear for which we have 2 ways.Actually we can make them sit together(at the rear)(consecutively)either starting from the left of the sedan or from the right of the sedan.But again for each way,there exits 2 other ways when they will change their positions.For example let the sisters be s1 and s2.If the sit from the left side,it could be either s1s2 or s2s1.Similarly from the right.Thus in all we get 2*2 ways.
3.Left is now 2 persons which can be made to sit in 2*1 ways.
4.In all,the sisters can be made to sit in 2*2*2*2=2*4*2=16 ways.
5.The answer now is 48-16=32.

I hope that the answer is correct and you are satisfied with the solution.
If not then please let me know what the right answer is.
THANK YOU AND GOOD LUCK.

[JonathanSchneider]

This problem has been discussed a couple of times on our forums before. Check this link for one discussion:
viewtopic.php?f=32&t=3037&p=10543&hilit=sedan#p10543

To view more on this problem, just search for "sedan" in the forums history (use "Search" up top).