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# A family with 1 mother, 1 father, and 3 children sit in a car with 2

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Math Expert
Joined: 02 Sep 2009
Posts: 51258
A family with 1 mother, 1 father, and 3 children sit in a car with 2  [#permalink]

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30 May 2016, 12:20
00:00

Difficulty:

45% (medium)

Question Stats:

54% (01:03) correct 46% (00:43) wrong based on 200 sessions

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A family with 1 mother, 1 father, and 3 children sit in a car with 2 front seats and 3 back seats. If the mother drives with a child in the front, and the father sits in the back with the other two children, how many seating arrangements are possible?

(A) 3
(B) 4
(C) 6
(D) 12
(E) 18

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A family with 1 mother, 1 father, and 3 children sit in a car with 2  [#permalink]

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30 May 2016, 13:45
I wasn't sure if the order of the way children are arranged matters in this question. Since children were mentioned as one (imagine we have 1 RED marble, 1 BLUE and 3 GREEN) and the front two seats are already determined, then there are 3 ways how the father can be seated in the back seat with regards to his children:

F C C
C F C
C C F

But if the order matters, then my solution doesn't hold. I am looking forward to discover the actual solution!
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Re: A family with 1 mother, 1 father, and 3 children sit in a car with 2  [#permalink]

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30 May 2016, 16:58
In the front seat we have 3 possibilities of children. (C1 or C2 or C3)
In the back seat we have (2!*1)*3 = 6 possibilities ( C1,C2,F / C2,C1,F / C1,F,C2 / C2,F,C1 / F,C1,C2 / F,C2,C1 -> IF C3 is in front seat)
So, 6*3 = 18 (try to figure out to change the possibilities of sitting back the C3 child in the place of C1 or C2 and do the 6 possibilities again and again).

Or, another way to see:
Looking into the back seat:
( C1,C2,F / C2,C1,F / C1,F,C2 / C2,F,C1 / F,C1,C2 / F,C2,C1) = 6
or +
( C3,C2,F / C2,C3,F / C3,F,C2 / C2,F,C3 / F,C3,C2 / F,C2,C3) = 6
or +
( C1,C3,F / C3,C1,F / C1,F,C3 / C3,F,C1 / F,C1,C3 / F,C3,C1) = 6
-------
18
Letter E
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Re: A family with 1 mother, 1 father, and 3 children sit in a car with 2  [#permalink]

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31 May 2016, 04:32
1
Seat of mother is fixed
one child is in the front but we don't know which one.
So lets consider child one is in the front, so we have 3! = 6 arrangements in the backseat
then consider child two is in the front, again we have 3! = 6 arrangements
And the same for the last one. 6*3 = 18, hence E
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Posts: 7108
Re: A family with 1 mother, 1 father, and 3 children sit in a car with 2  [#permalink]

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31 May 2016, 05:04
1
fantaisie wrote:
I wasn't sure if the order of the way children are arranged matters in this question. Since children were mentioned as one (imagine we have 1 RED marble, 1 BLUE and 3 GREEN) and the front two seats are already determined, then there are 3 ways how the father can be seated in the back seat with regards to his children:

F C C
C F C
C C F

But if the order matters, then my solution doesn't hold. I am looking forward to discover the actual solution!

Hi,

But here the order of chidren will also matter as these are different identities..
so 3*3!... 3! is the way these 3 kids can be arranged within themselves
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Re: A family with 1 mother, 1 father, and 3 children sit in a car with 2  [#permalink]

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12 Jun 2017, 02:38
[quote="chetan2u"]

If the question asks about number of arrangements then why are we not considering the mothers seat combination ?

We got 18 but there are 3 combinations of either of 3 child sitting with the mother, Then we the answer is not 21?
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Re: A family with 1 mother, 1 father, and 3 children sit in a car with 2  [#permalink]

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14 Jun 2017, 15:20
1
Bunuel wrote:
A family with 1 mother, 1 father, and 3 children sit in a car with 2 front seats and 3 back seats. If the mother drives with a child in the front, and the father sits in the back with the other two children, how many seating arrangements are possible?

(A) 3
(B) 4
(C) 6
(D) 12
(E) 18

We can let F be the father, M be the mother, and A, B, and C be the 3 children. We are given that the mother must occupy the driver’s seat and one of the children is sitting in the front.

So, let’s assume that M and A are sitting in the front; this leaves F, B, and C sitting in the back. The front row seats are fixed since M must be at the driver’s seat, but the back row seats are not. If F, B, and C are sitting in the back, there are 3! = 6 ways to occupy the back row seats.

Similarly, if M and B are sitting in the front and F, A, and C are sitting in the back, there will be also 6 ways to occupy the back row seats. Lastly, if M and C are sitting in the front and F, A, and B are sitting in the back, there will be another 6 ways to occupy the back row seats.

Thus the total number of seating arrangements is 6 + 6 + 6 = 18.

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Re: A family with 1 mother, 1 father, and 3 children sit in a car with 2  [#permalink]

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12 Nov 2018, 12:12
In the back seat we can pic two children in 3C2 ways and F C C can be arranged in 3! ways.
So, 3C2 x 3! ways.
Since the child can not drive so, C M can be arranged in one one way.
so ans is 3C2 X 3!=18
Re: A family with 1 mother, 1 father, and 3 children sit in a car with 2 &nbs [#permalink] 12 Nov 2018, 12:12
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