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Math Expert V
Joined: 02 Sep 2009
Posts: 61524

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11 00:00

Difficulty:   35% (medium)

Question Stats: 73% (01:36) correct 27% (02:31) wrong based on 83 sessions

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If among 5 children there are 2 siblings, in how many ways can the children be seated in a row so that the siblings do not sit together?

A. 38
B. 46
C. 72
D. 86
E. 102

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Math Expert V
Joined: 02 Sep 2009
Posts: 61524

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Official Solution:

If among 5 children there are 2 siblings, in how many ways can the children be seated in a row so that the siblings do not sit together?

A. 38
B. 46
C. 72
D. 86
E. 102

Without limitations, 5 children can be seated in $$5! = 120$$ ways. Find the number of ways to seat the 5 children so that the siblings DO sit together. The siblings can be regarded as one unit so there are 4! combinations. But within this unit the siblings can sit in two different ways. So the number of ways to seat the 5 children so that the siblings DO sit together is $$4!*2 = 48$$. Thus, the number of combinations in which the siblings DO NOT sit together is 120 - 48 = 72.

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Joined: 18 Aug 2014
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Bunuel wrote:

The siblings can be regarded as one unit so there are 4! combinations. But within this unit the siblings can sit in two different ways. So the number of ways to seat the 5 children so that the siblings DO sit together is $$4!*2 = 48$$.

I thought 4! included all the possibilities not just a particular order? So that there is no need to multiply the 4! by 2.

Like how many ways can you arrange a, b, and c

3! = 6, as below shows if B and C were siblings both placements are accounted for are they not?

A B C
A C B
B A C
B C A
C A B
C B A
Math Expert V
Joined: 02 Aug 2009
Posts: 8256

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redfield wrote:
Bunuel wrote:

The siblings can be regarded as one unit so there are 4! combinations. But within this unit the siblings can sit in two different ways. So the number of ways to seat the 5 children so that the siblings DO sit together is $$4!*2 = 48$$.

I thought 4! included all the possibilities not just a particular order? So that there is no need to multiply the 4! by 2.

Like how many ways can you arrange a, b, and c

3! = 6, as below shows if B and C were siblings both placements are accounted for are they not?

A B C
A C B
B A C
B C A
C A B
C B A

Hi,
the 2 is for the two siblings who have been taken as 1..

The siblings can be regarded as one unit so there are 4! combinations..
Let these siblings be $$S_1$$ and $$S_2$$ AND others be A,B and C..
so one of the arrangements out of 4! is $$A,B,C,S_1,S_2$$...
But we can arrange the siblings in two ways here..
second would be $$A,B,C,S_2,S_1$$...

Similarily there will be two ways for each ARRANGEMENT..
that is why 4!*2..
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On the other hand if we don't see the cunning plan of treating the siblings as one unit, we could do as follows:

Sibling 1 and Sibling 2 occupy 2 spaces out of 5, there are 4 ways for them to sit together, in each of these 4 ways they can switch position>
S1,S2,A,B,C; S2,S1,A,B,C>> the 3 children that are not siblings are arranged as follows: 3! (since the kids occupy the 3 lest over spaces). the 2 siblings can be arranged in 2 ways giving us a total of 3*2*2=12 ways to sit the children for the first arrangement.. that includes S1,S2,B,C,D.. S2,S1,A,C,B (for example)... If you do the same calculus for all the positions where the siblings sit close (so.. A,S1,S2,C,B... A,B,S1,S2,C.. A,B,C,S1,S2) the result is 48.

Total arrangements possible : 5! - 48= 120-48= 72

I hope it helps,
best regards.
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Joined: 11 Nov 2014
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I applied different approach -

there are 5 seats , 2 siblings S1 and S2 and three others A,B,C

________ ________________ _____________ __________ ______________
---5 ways ----------3 ways----------- 3 ways--------2 ways------1 way
------------ (not counting one sibling)
5*3*3*2*1 = 90
order of seating can be in any way.
can you please tell me where am I wrong ? Bunuel
Intern  Joined: 01 Sep 2016
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Simpler way to do this: We have 3 children and two siblings

Arrange 3 children in $$3!$$ ways

Consider the remaining two children: they have 4 possible places to be placed (ie, anywhere in _, where X represents the other 3 children): _ X _ X _ X _

Hence we can arrange the two siblings in 4P2 ways = $$\frac{4!}{(4 - 2)!}$$

Hence total probability is $$3! * \frac{4!}{(4 - 2)!}$$ = $$(3*2* 1) * \frac{(4*3*2*1)}{(2*1)}$$ = $$6 * 12$$ = $$72$$

This can be generalized for N children & X siblings
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Hi Bunuel ,
I got your explanation. But i tried with below approach,and answer was wrong.Could you please clarify where i was wrong

No of way to arrange 5 where 2 are one kind 5!/2!
Total no of arrangement for 5 =5!

Thanks
Math Expert V
Joined: 02 Sep 2009
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abhinashgc wrote:
Hi Bunuel ,
I got your explanation. But i tried with below approach,and answer was wrong.Could you please clarify where i was wrong

No of way to arrange 5 where 2 are one kind 5!/2!
Total no of arrangement for 5 =5!

Thanks

How can you consider siblings to be identical and indistinguishable?

Yes, the number of ways to arrange AABCD is 5!/2! but how does this give the arrangements when A's are together? The arrangements will be:
AABCD
ABACD
ABCDA
...
BCDAA
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Manager  S
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ABCSS

Let S sits first----- for the 2nd chair we have 3 choices. For the 3rd chair we have 3 and for the 4th chair we have 2 choices = 1*3*3*2*1 = 18

Now we can Put first S in any 4 chairs. So we have 4*18 = 72 choices
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Bunuel wrote:
If among 5 children there are 2 siblings, in how many ways can the children be seated in a row so that the siblings do not sit together?

A. 38
B. 46
C. 72
D. 86
E. 102

1. "seated in a row" = order matters, permutation
2. replacement? No
3. different objects? Yes
4. different groups? No (row = 1 group)

Slot method:
A1 first slot = 3
A1 _ A2 _ _
A1 _ _ A2 _
A1 _ _ _ A2

A1 2nd slot = 2
_ A1 _ A2 _
_ A1 _ _ A2

A1 3rd slot = 1
_ _ A1 _ A2

(3+2+1) or 3P1 = 6 * 2 (doubled for A2) = 12 ways to arrange the 2 siblings
3P1 = 3*2*1 ways to arrange the other children
12*6 = 72

Another way to think about it ... ways*choices:

Step 1) Choices of objects per slot, 2P1 for siblings * 3P1 for others = 12
_2_ * _3_ * _1_* _2_* _1_ = 12

Step 2) Ways to position the siblings among 5 slots so they don't sit together, 3C1 * 2 = 6
(leftmost slot, 2nd to leftmost = 2 and middle are viable --> 3 * 2 for each sibling = 6 )

ways * choices = 6 ways * 12 obj choices * 1 grp choice = 72
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Might be a good idea to add '96' as one of the options. I would've picked that if it were there, because I forgot to multiply by 2 to account for AB and BA as different seating configurations for the siblings. Re: M08-30   [#permalink] 30 Nov 2019, 06:54
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# M08-30

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