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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]

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Updated on: 27 Apr 2012, 03:07
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Difficulty:

55% (hard)

Question Stats:

66% (02:17) correct 34% (02:34) wrong based on 674 sessions

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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

Originally posted by pratikbais on 27 Apr 2012, 02:57.
Last edited by Bunuel on 27 Apr 2012, 03:07, edited 1 time in total.
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A family consisting of one mother, one father, two daughters  [#permalink]

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27 Apr 2012, 03:07
37
21
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.

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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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28 Dec 2012, 19:05
9
3
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.

$$=4*2*2*2 = 32$$

##### General Discussion
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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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11 Sep 2017, 14:56
7
1
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

We can analyze the problem using the following two cases: 1) the father is the driver and 2) the mother is the driver.

Case 1: The father is the driver.

If the father is the driver, we could have the following subcases:

i) The mother sits in the front row beside the father. Since the daughters refuse to sit next to each other, there are 2 seating arrangements in the back row: Dsd, dsD (D = elder daughter, d = younger daughter, and s = son).

ii) The son sits in the front row beside the father. Since the daughters refuse to sit next to each other, there are 2 seating arrangements in the back row: Dmd, dmD (m = mother).

iii) The elder daughter sits in the front row beside the father. Since, in this case, the two daughters will definitely not sit next to each other, there are 3! = 6 seating arrangements for the 3 remaining people who sit in the back row.

iv) The younger daughter sits in the front row beside the father. Like subcase (iii), there will be 6 seating arrangements in the back row.

As we can see from the above, if the father is the driver, there will be a total of 2 + 2 + 6 + 6 = 16 seating arrangements. We can make the same argument when the mother is the driver. Thus, there will be another 16 seating arrangements, and hence we have a total of 16 + 16 = 32 seating arrangements.

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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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27 Apr 2012, 09:30
3
3
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]

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08 Sep 2017, 04:28
3
1
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

2C1- Parent for driving seat

Case 1: one daughter sits on front seat
2C1- to select front seat daughter
3! - arrangement of rest 3 on back seats

Case 2: Both sisters sit on back seat at extreme corners
2! - arrangement of sisters
2! - Arrangement of one parent and the boy

total outcomes = 2C1*(2C1*3!+2!*2!)
= 2*(12+4) = 32

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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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26 Jul 2014, 09:23
2
1
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)

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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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16 Sep 2014, 12:34
1
alphonsa wrote:
Bunuel 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

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.

Sorry Bunuel, but i didnt get the above in red.
Why the 4*2*2*1

4 ways to sit sisters together;
2 ways to fill drivers seat (mother or father);
2 ways to fill front seat (3 people are already distributed, so 2 are left);
1 way to fill the remaining back seat.

Total = 4*2*2*1.

Hope it's clear.
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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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17 Aug 2018, 18:10
1
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

Official Solution (Credit: Manhattan Prep)

The easiest way to solve this question is to consider the restrictions separately. Let’s start by considering the restriction that one of the parents must drive, temporarily ignoring the restriction that the two sisters won't sit next to each other.

This means that…
2 people (mother or father) could sit in the driver’s seat
4 people (remaining parent or one of the children) could sit in the front passenger seat
3 people could sit in the first back seat
2 people could sit in the second back seat
1 person could sit in the remaining back seat

The total number of possible seating arrangements would be the product of these various possibilities: 2 × 4 × 3 × 2 × 1 = 48

We must subtract from these 48 possible seating arrangements the number of seating arrangements in which the daughters are sitting together. The only way for the daughters to sit next to each other is if they are both sitting in the back.

This means that…
2 people (mother or father) could sit in the driver’s seat
2 people (remaining parent or son) could sit in the front passenger seat

Now for the back three seats we will do something a little different. The back three seats must contain the two daughters and the remaining person (son or parent). To find out the number of arrangements in which the daughters are sitting adjacent, let’s consider the two daughters as one unit. The remaining person (son or parent) is the other unit. Now, instead of three seats to fill, we only have two "seats," or units, to fill.
There are 2 × 1 = 2 ways to seat these two units.
However, the daughter-daughter unit could be d1d2 or d2d1
We must consider both of these possibilities so we multiply the 2 by 2! for a total of 4 seating possibilities in the back.
We could also have manually counted these possibilities:
d1d2X, d2d1X, Xd1d2, Xd2d1

Now we must multiply these 4 back seat scenarios by the front seat scenarios we calculated earlier:
(2 × 2) × 4 = 16
front back

If we subtract these 16 "daughters-sitting-adjacent" scenarios from the total number of "parent-driving" scenarios, we get: 48 – 16 = 32

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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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26 Jul 2014, 09:53
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)

_______________
Yes, that's correct.
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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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16 Sep 2014, 02:25
Bunuel 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

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.

Sorry Bunuel, but i didnt get the above in red.
Why the 4*2*2*1
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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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22 May 2017, 19:09
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

This problem can be tackled best via a direct method.

There are two options for drivers (mother or father) = 2.

Let's assume one of the sisters takes the passenger seat in the car. Then the number of spots for the rest of the family members is 3! = 3*2

2*3*2 = 12 ways * 2 sisters who could each sit in the front = 24 options.

Now let's assume they both sit in the back seat. There are 2 options for the driver, 2 options for the passenger, and only one option for the back seat (the person separating the sisters) 2*2*1 = 4 * 2 sisters = 8 options.
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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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03 Feb 2018, 15:54
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 front… = 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

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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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18 Dec 2018, 22:18
There are two possibilities of arrangement.
Case 1. When 1 daughter is in front.
Total ways = 2(no of ways to choose driver) x 2(no of ways to choose 1 daughter for front seat) x 3!(arrangement of 3 people in back seats. = 2 x 2 x 6 = 24 ways.
Case 2. When both daughters are sitting in the back and one person is sitting between them.
Total ways = 2(ways to choose driver) x 2( ways to choose person to be seated in front) x 2(arrangement of two daughters in the left and right of person sitting in the back seat)= 8 ways.
So, total no of arrangements = 32.
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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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23 Nov 2019, 09:19
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

considering parent taking driver seat = 2c1
and daughter front seat = 1
last 3 seats can be occupied in 3! ways
case 2:
3 seats occupied by 2 daughters ; 2c1 and 1 seat among son and 1 parent ; 2c1
total ways ; 2*(2c1*3!+2c1*2c1 ) = 32
IMO B
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A family consisting of one mother, one father, two daughters  [#permalink]

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23 May 2020, 12:05
Bunuel 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

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.

Bunuel, i was doing 5!-16
why are we considering the one condition as driver to be either mother/father while taking total arrangement- arrangement of sister sitting together
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Re: A family consisting of one mother, one father, two daughters  [#permalink]

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23 May 2020, 19:13
kunalc20 wrote:
Bunuel 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

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.

Bunuel, i was doing 5!-16
why are we considering the one condition as driver to be either mother/father while taking total arrangement- arrangement of sister sitting together

Hi kunalc20,

If you want to start with 5! (re: the assumption that any person could sit in any seat), and then 'subtract out' all of the possibilities that DON'T fit the various 'seating rules', then that's fine, but you would have to remove far more than just 16 of the 120 possibilities. ONLY the Mother or Father can sit in the driver's seat, so that alone removes (3)(4!) = 72 of the options (these are the options in which one of the three CHILDREN is sitting in the driver's seat). Once you've dealt with that issue - and you have 48 options remaining - you then also have to remove the options in which the Mother or Father is driving AND the 2 sisters are sitting next to one another in the back seat.

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Re: A family consisting of one mother, one father, two daughters   [#permalink] 23 May 2020, 19:13