seating. source: manhattan

[jsasipriya]

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?
A 28
B 32
C 48
D 60
E 120

OA B

If you know a simple method to determine this, please share.

Thanks

[kvcpk]

MFSD1D2
driver seat can be filled in 2 ways.

Assume Pillion seat is filled by the other parent. this can be done in 1 way
Then, back seats can be filled in only 2 ways (D1SD2 or D2SD1) so that the daughters dont sit together.
All three should happen together. Hence 2*1*2= 4 ways.

Assume Pillion seat is filled by the son. this can be done in 1 way
Then, back seats can be filled in only 2 ways (D1MD2 or D2MD1) so that the daughters dont sit together.
All three should happen together. Hence 2*1*2 = 4 ways.

Assume Pillion seat is filled by the daughter. this can be done in 2 ways.
Then, back seats can be filled in only 3! = 6 ways because there is no restriction.
All three should happen together. Hence 2*2*6 = 24 ways.

Any of these should occur. Hence Add. 4+4+24 = 32 ways.

Hope this helps!!

[indiantiger]

D-------P

R1 R2 R3

D - driver seat
P - passenger seat
R1 = rear 1
R2 = rear 2
R3 = rear 3

Case 1 :

D can be filled in 2 ways with either parent
P can be filled in 2 ways with either parent not driving or son
R1 can be filled in 2 ways by either sister
R2 has to filled by who ever did not get to sit on P (either son or parent) 1 way
R3 can be filled by left out sister.

total ways in this case = 8

Case 2
D can be filled in 2 ways with either parent
P can be filled in 2 ways with either sister
R1 can be filled in 3 ways by sister who did not get the P seat, son or parent who is not driving
R2 has to filled 2 ways by (sister who did not get the P seat, son or parent who is not driving) -1 as one of these must have sat on R1
R3 can be filled by left out person
total ways in this case = 24

Total overall cases = 24+8 = 32.