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A family consisting of one mother, one father, two daughters

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A family consisting of one mother, one father, two daughters [#permalink] New post 27 Apr 2012, 02:57
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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
[Reveal] Spoiler: OA

Last edited by Bunuel on 27 Apr 2012, 03:07, edited 1 time in total.
Added the answer choices and the OA
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Re: Seating arrangements [#permalink] New post 27 Apr 2012, 03:07
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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

Approach #1:

Sisters can sit separately in two ways:

1. one of them is on the front seat (2 ways). Others (including second sister) can be arranged in: 2 (drivers seat)*3! (arrangements of three on the back seat)=12 ways. Total for this case: 2*12=24

Or

2. both by the window on the back seat (2 ways). Others can be arranged in: 2 (drivers seat)*2 (front seat)*1(one left to sit between the sisters on the back seat)=4 ways. Total for this case: 2*4=8.

Total=24+8=32.

Approach #2:

Total # of arrangements:
Drivers seat: 2 (either mother or father);
Front seat: 4 (any of 4 family members left);
Back seat: 3! (arranging other 3 family members on the back seat);
So. total # of arrangements is 2*4*3!=48.

# of arrangements with sisters sitting together:
Sisters can sit together only on the back seat either by the left window or by the right window - 2, and either {S1,S2} or {S2,S1} - 2 --> 2*2=4;
Drivers seat: 2 (either mother or father);
Front seat: 2 (5 - 2 sisters on back seat - 1 driver = 2);
Back seat with sisters: 1 (the last family member left);
So, # of arrangements with sisters sitting together is 4*2*2*1=16.

48-16=32.

Answer: B.
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Re: A family consisting of one mother, one father, two daughters [#permalink] New post 27 Apr 2012, 09:30
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pratikbais 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


Steps to solve this problem -
People - 1M, 1F, 1S, 2D (D1,D2)
1. Choose a parent to drive the sedan.
2. Find the ways to seat the daughters.
3. Place the remaining 3 family members.
4. Finally multiply results from steps 1, 2 and 3.

1. Ways to choose a parent to drive = 2 (One person seated, total remaining = 4).
2. Arrangement in which daughters sit separate = Total Arrangements of 4 people - Arrangements with D1 and D2 glued together.

-> 4! - 4*2*1

4*2*1 - the pair of daughters can take 2 out of 3 consecutive spots at the back seat. Also they can interchange seats D1D2 and D2D1 are different arrangements. So total 4.
Remaining two people sit in 2*1 ways.

(This takes care of step 3 as well)

Finally step 4 - Total = 2* (4! - 4*2*1 ) = 2* (24-8) = 2*16 = 32.

Thanks.
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Re: A family consisting of one mother, one father, two daughters [#permalink] New post 28 Dec 2012, 19:05
pratikbais 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


Case 1: Both daughters in the back seat but seated separately
Case 2: One daughter in front seat and the other in the middle of the back seat
Case 3: One daughter in front seat and the other in the right back seat
Case 4: One daugther in front seat and the other in the left back seat

That's 4 positions and two daughters are interchangeable. Thus, 8.

There are 2 ways to select a parent and 2 ways to seat the son and another parent.

=4*2*2*2 = 32

Answer: B
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Re: A family consisting of one mother, one father, two daughters [#permalink] New post 24 Jul 2014, 21:21
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Re: A family consisting of one mother, one father, two daughters [#permalink] New post 26 Jul 2014, 09:23
Could someone tell me is my approach is correct?

1) Total of arrangement in the driver seat : 2 (mother or father)

2) Total arrangement of the four others : 4!

3) Total ways with the 2 daughters next to each other : 4 (left window D1-D2 and D2-D1, right window D1-D2 and D2-D1) *2 (we have to place the mother or father that is not driving)

Answer = 2*(4!-4*2) = 32
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Re: A family consisting of one mother, one father, two daughters [#permalink] New post 26 Jul 2014, 09:53
Expert's post
oss198 wrote:
Could someone tell me is my approach is correct?

1) Total of arrangement in the driver seat : 2 (mother or father)

2) Total arrangement of the four others : 4!

3) Total ways with the 2 daughters next to each other : 4 (left window D1-D2 and D2-D1, right window D1-D2 and D2-D1) *2 (we have to place the mother or father that is not driving)

Answer = 2*(4!-4*2) = 32

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Yes, that's correct.
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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: A family consisting of one mother, one father, two daughters   [#permalink] 26 Jul 2014, 09:53
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