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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## A family consisting of one mother... tagged by: Brent@GMATPrepNow ##### This topic has 3 expert replies and 5 member replies ## A family consisting of one mother... 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 Source: MGMAT CAT EXAM Answer: B I have no idea where to start here!! Help!! Thanks!! Legendary Member Joined 15 Apr 2011 Posted: 1085 messages Followed by: 21 members Upvotes: 158 It's simple indeed Let's consider, Farther is driving first and the pertinent cases 1 daughter sits in the front seat and others seat in the back seats 3!. Two such cases 2*3! Son is in the front seat, mom is in the middle, 2 ways. Mom is in the front seat and Son is in the middle 2 ways. Total makes 2*3!+2+2=16 ways. The same number of ways when Mom is driving and Farther takes his turn 16*2=32 b factor26 wrote: 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 Source: MGMAT CAT EXAM Answer: B I have no idea where to start here!! Help!! Thanks!! _________________ Success doesn't come overnight! Legendary Member Joined 16 Oct 2011 Posted: 588 messages Followed by: 9 members Upvotes: 130 Test Date: 3rd May '12 Target GMAT Score: 750+ GMAT Score: 720 factor26 wrote: 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 Source: MGMAT CAT EXAM Answer: B I have no idea where to start here!! Help!! Thanks!! You can start of bu drawing the kind of arrangements 2 in the front 3 at the back Driver has to be a parent, so we have 2 options for one of the front seats Quote: the two daughters refuse to sit next to each other, Lets say one of the sisters sit at the front seat So we have 2 options for the second seat in the front and 3 people to arrange on 3 seats at the back Arrangements = 2*2*3*2*1 = 24 Now lets say both the sisters sit on the backseats NO of options for the front seat= 2 [son or one of the parent] There has to be one person b/w the two sisters the arrangements would be like this No of arrangements= 8 Total no of arrangements = 24+8 = 32 Option B ### GMAT/MBA Expert GMAT Instructor Joined 25 May 2010 Posted: 15257 messages Followed by: 1861 members Upvotes: 13060 GMAT Score: 790 _________________ Mitch Hunt Private Tutor for the GMAT and GRE GMATGuruNY@gmail.com If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon. Available for tutoring in NYC and long-distance. For more information, please email me at GMATGuruNY@gmail.com. Student Review #1 Student Review #2 Student Review #3 Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now. Newbie | Next Rank: 10 Posts Joined 08 Jul 2012 Posted: 6 messages GuruNY, I have tried a different approach. Not sure, if I got the answer by fluke or the approach is correct. Please help me understand if my logic is correct. Thanks! Driver seat=2 Rest four seats=4! Two sisters always sit together = considering 2 sisters as 1 unit = 4c3 *2! Solution = 2(4!-4c3*2!)=2*16=32 GMATGuruNY wrote: ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 12833 messages Followed by: 1247 members Upvotes: 5254 GMAT Score: 770 factor26 wrote: 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 ! Here's another approach. It's a little longer, but it demonstrates another way to tackle restrictions. The restriction about the sisters is somewhat problematic, so I decided to ignore the rule and seat all 5 people without obeying that restriction. Then once I determine the total number of arrangements, I subtract the number of arrangements where the sisters are sitting together. Number the seats as follows: Seat #1: driver's seat Seat #2: passenger's seat Seats #3, 4, 5: back seats # of arrangements where we ignore rule about the sisters not sitting together Take the task of seating all 5 people and break into stages. Stage 1: Seat someone in seat #1 Only a parent can sit here. So, this stage can be accomplished in 2 ways. Stage 2: Seat someone in seat #2 Once we have seated someone in seat #1, there are 4 people remaining. So, this stage can be accomplished in 4 ways. Stage 3: Seat someone in seat #3 At this point, we have already seated 2 people, so there are now 3 people remaining. So, this stage can be accomplished in 3 ways. Stage 4: Seat someone in seat #4 There are 2 people remaining, so this stage can be accomplished in 2 ways. Stage 5: Seat someone in seat$5
This stage can be accomplished in 1 way

By the Fundamental Counting Principle (FCP) we can complete all 5 stages (and thus seat all 5 people) in (2)(4)(3)(2)(1) ways
48 ways

So there are 48 different ways to seat the family such that a parent drives. At this point, the 48 different arrangements include arrangements where the sisters are seated together. So, we need to SUBTRACT the number of arrangements where the sisters are seated together.

There are two cases where the sisters are together.
case 1: the sisters are in seats #3 and #4
case 2: the sisters are in seats #4 and #5

case 1: the sisters are in seats #3 and #4
Once again, we'll take the task of seating everyone and break it into stages:
Stage 1: seat a parent in seat #1.
Must be 1 of 2 parents. So, this stage can be accomplished in 2 ways.
Stage 2: seat a sister in seat #3
Must be 1 of 2 sisters. So, this stage can be accomplished in 2 ways.
Stage 3: seat the other sister in seat #4.
Once we have seated a sister in seat #3, only 1 sister remains. So, this stage can be accomplished in 1 way.
Stage 4: seat someone in seat #2.
At this point, we have seated 3 people, so only 2 people remain. So, this stage can be accomplished in 2 ways.
Stage 5: seat someone in seat #5.
One person remaining. So, this stage can be accomplished in 1 way.

By the Fundamental Counting Principle (FCP) we can complete all 5 stages (and thus seat the sisters are in seats #3 and #4) in (2)(2)(1)(2)(1) ways
8 ways

case 2: the sisters are in seats #4 and #5
We can follow the same steps as above to get 8 more arrangements

So the final answer is 48 - 8 - 8 = 32

Cheers,
Brent

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Newbie | Next Rank: 10 Posts
Joined
28 May 2013
Posted:
7 messages
Can anybody please help me with this ... I am not able to understand the whole. If I try to apply alphabets order.
Assume Father stands for F , Mother for M , Son for S and Daughters for D.
Now taking the required arrangement of the in first line we can have F/M
in second line any of M/F,D,S,D where D can not come together.
so the possible arrangements for first line : 2!
for second (4! - 4C3 *2) (where two daughters combined to one count)
my concern Is it a right approach?

Thanks

Senior | Next Rank: 100 Posts
Joined
17 Sep 2015
Posted:
57 messages
Brent,

How about the way I have done.

1)Seat 1: Either parent seats: 2 ways
2)Seat 3:Any of the 4 seats: 4 ways
3)Seat 4:Any of the pending 3 except sister sits: 2 ways
4)Seat 2:Pending 2 sits: 2 ways
5)Seat 5:Pending 1 sits: 1 way

Answer 32. Correct me if my understanding and method is wrong.

### GMAT/MBA Expert

Elite Legendary Member
Joined
23 Jun 2013
Posted:
10130 messages
Followed by:
494 members
2867
GMAT Score:
800
Hi All,

These types of questions can be approached a couple of different ways. There's a "visual" aspect to this question that can help you to take advantage of some shortcuts built into the prompt, so I'm going to use a bit of "brute force" and some pictures to answer this question. Since we're arranging people in seats, we'll end up doing some "permutation math."

M = Mother
F = Father
D1 = 1st Daughter
D2 = 2nd Daughter
S = Son

Front Back
_ _ _ _ _
1st spot = driver

We have 2 restrictions that we have to follow:
1) Either the Father or Mother must be the driver
2) The two daughters CANNOT sit next to one another

Let's put the Mother in the driver's seat and count up the possibilities:

M F (2)(1)(1) Here, the two daughters have to be separated by the son, but either daughter could be in the "first back seat" = 2 options

M D1 (3)(2)(1) Here, with the first daughter up front, the remaining 3 people (F, D2 and S) can be in any of the back seats = 6 options
M D2 (3)(2)(1) Here, we have the same situation, but with the second daughter up frontâ€¦ = 6 options
M S (2)(1)(1) Here, with the son up front, we have the same scenario as we had when the Father was up front = 2 options

Total options with Mother driving = 2+6+6+2 = 16 options

Now we can take advantage of the shortcut I mentioned earlier - We can flip-flop the Mother and Father in the above examples. This will gives us another 16 options with the Father driving.

Total options: 16 + 16 = 32 options

GMAT assassins aren't born, they're made,
Rich

_________________
Contact Rich at Rich.C@empowergmat.com

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