Problem Solving

I'm having trouble figuring out the best way to account for the two daughters not being able to sit next to each other. Any ideas?


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

IMO B

Let P --> Parents, G--> daughter and B -->Son

Case 1 when Both Parents are seating in front and kids at the back such that both daughters don't sit next to eachother

Front---> PP ---->2! ways (Driver's seat can be shared by any mom and dad so 2 ways)
Back---->GBG ---> 2! (girls can switch sides so 2 ways)

Total 2*2 = 4 ways

Case 2 when the son is in the front seat and one of the parents at the middle back seat

Front --> PB ---> 2! ways (Driver's seat can be shared by any mom and dad so 2 ways)
Back--> GPG ---> 2! (girls can switch sides so 2 ways)

Total 2*2 = 4 ways

Case 3 when one of the Girls is in the front seat
Front --> PG --> 2*2 =4 ways (parents and girls can switch their seats)
Back --> GPB--> 3! = 6 ways

Total 4*6 = 24 ways

Combining all three cases 4+4+24 = 32 ways