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Two couples and one single person are seated at random in a [#permalink]
25 Sep 2009, 13:07

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Difficulty:

75% (hard)

Question Stats:

46% (02:55) correct
54% (01:43) wrong based on 28 sessions

Two couples and one single person are seated at random in a row of five chairs. What is the probability that neither of the couples sits together in adjacent chairs?

Let's call the first couple C and c, the second couple K and k, and the single person S. Let's seat S in different places and figure out the possible ways to have no couples sit together. If S sits in the first seat, any of the remaining four people could sit next to S. However, only two people could sit in the next seat: the two who don't form a couple with the person just seated). For example, if we have S K so far, C or c must sit in the third seat. Similarly, we have only one choice for the fourth seat: the remaining person who does not form a couple with the person in the third seat. Because we have seated four people already, there is only one choice for the fifth seat; the number of ways is 4 × 2 × 1 × 1 = 8. Because of symmetry, there are also 8 ways if S sits in the fifth seat. Now let's put S in the second seat. Any of the remaining four could sit in the first seat. It may appear that any of the remaining three could sit in the third seat, but we have to be careful not to leave a couple for seats four and five. For example, if we have C S c so far, K and k must sit together, which we don't want. So there are only two possibilities for the third seat. As above, there is only one choice each for the fourth and fifth seats. Therefore, the number of ways is 4 × 2 × 1 × 1 = 8. Because of symmetry, there are also 8 ways if S sits in the fourth seat. This brings us to S in the third seat. Any of the remaining four can sit in the first seat. Two people could sit in the second seat (again, the two who don't form a couple with the person in the first seat). Once we get to the fourth seat, there are no restrictions. We have two choices for the fourth seat and one choice remaining for the fifth seat. Therefore, the number of ways is 4 × 2 × 2 × 1 = 16. We have found a total of 8 + 8 + 8 + 8 + 16 = 48 ways to seat the five people with no couples together; there is an overall total of 5! = 120 ways to seat the five people, so the probability is >>>>> 48/120 or 2/5 <<<<<

Re: very hard probability / combination (700+) helpppp [#permalink]
25 Sep 2009, 15:45

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Expert's post

I posted a solution here, which is essentially the same type of case analysis as in the above. There's another solution in the thread below that reverses the problem: count all the arrangements possible, then subtract all the arrangements in which at least one couple does sit together.

I don't see a ten-second solution, however. _________________

Nov 2011: After years of development, I am now making my advanced Quant books and high-level problem sets available for sale. Contact me at ianstewartgmat at gmail.com for details.

Re: very hard probability / combination (700+) helpppp [#permalink]
25 Sep 2009, 20:40

IanStewart wrote:

I posted a solution here, which is essentially the same type of case analysis as in the above. There's another solution in the thread below that reverses the problem: count all the arrangements possible, then subtract all the arrangements in which at least one couple does sit together.

Re: very hard probability / combination (700+) helpppp [#permalink]
25 Sep 2009, 22:03

I too solved the prob the other way round. The total number of ways that none sit together = 5! - (Total number of ways in which atleast one pair sits together) = 5! - 72 = 48

Re: very hard probability / combination (700+) helpppp [#permalink]
25 Sep 2009, 23:12

I got 2/5 as well, but I did it in a slightly different way. Maybe this approach is easier for some people, although I find Ian's explanation to be quite clear as well.

Basically, you have 1 single person, and two couples (couple A and B). So 5 people: A A B B and S. You want no repeating characters beside each other.

They sit in 5 chairs: _ _ _ _ _

3 Cases (based on where the single person is located)

Re: very hard probability / combination (700+) helpppp [#permalink]
22 Oct 2009, 01:37

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total outcomes 5! outcomes of couple A sit together: 2*4!, outcomes of couple B sit together: 2*4!, but we must deduct the possibility the 2 couples sit together: 3!*4 So outcomes of at least one couple sit together= 48+48-24=72 Outcomes of no couple sit together: 5!-72=48 possibility of no couple sit together: 48/5!=2/5

Re: very hard probability / combination (700+) helpppp [#permalink]
20 Oct 2011, 12:48

I have another way with just 2 cases to consider (with the same final answer of 2/5): For simplicity I'll rephrase the question as putting 2 A's, 2 B's, and 1 C in a row such that two A's are not adjacent and the two B's are not adjacent. _ _ _ _ _ 1 2 3 4 5

case 1: The letters on either end of the row (positions 1 and 5) are not the same: This is equal to having no adjacent letters the same in the following diagrams:

_ _ _ _ _ 1 2 3 4 5

_ _ _ _ _ 2 3 4 5 1

_ _ _ _ _ 3 4 5 1 2

_ _ _ _ _ 4 5 1 2 3

_ _ _ _ _ 5 1 2 3 4

I.e. the pairs 1 and 2, 2 and 3, 3 and 4, 4 and 5, 5 and 1 can not be the same.

If we place C on one of the 5 positions (this is 5 choices) we use the diagram which would have C in the middle. Now, we have only two possibilities for the position right of C , A or B:

_ _ C ? _

If ? is A, the only possibility is

A B C A B

likewise, if ? is B, the only possibility is

B A C B A

therefore there are only 2 total possibilities. 5 (places from c)*2 (possibilites per position of c) =10 total possibilities for the case.

case 2: The letters on positions 1 and 5 are the same: _ _ _ _ _ 1 2 3 4 5

The only 2 possibilites are

A B C B A

or

B A C A B

so 2 total possibilities for this case.

In conclusion:

10+2=12 total favourable possibilities

the number of ways to place 2A's, 2B's and a C is

5 (places for the C)*(4*3/2)(ways to place the two A's)=30. Note that after we placed the C and the A's, the B's have only a single way to be placed.

Re: very hard probability / combination (700+) helpppp [#permalink]
20 Oct 2011, 13:46

I used the glue method as referenced in Mgmat book

Total number of rearranging 5 ppl in 5 chairs = 5! = 120 (denominator)

Now lets resolve for the numerator: A1 A2 C B1 B2 treat A1 and A2 as a single person and B1 and B2 as a single person...then rearranging 3 ppl in 3 positions is 3! = 6

we need to account for 8 variations in the seating arrangements: A1 A2 C B1 B2 A1 A2 C B2 B1 A2 A1 C B1 B2 A2 A1 C B2 B1 B1 B2 C A1 A2 B1 B2 C A2 A1 B2 B1 C A1 A2 B2 B1 C A2 A1

Re: very hard probability / combination (700+) helpppp [#permalink]
20 Oct 2011, 22:10

2

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Expert's post

benjiboo wrote:

Two couples and one single person are seated at random in a row of five chairs. What is the probability that neither of the couples sits together in adjacent chairs?

1/5 1/4 3/8 2/5 1/2

I am giving my solution below which needs a little bit of thought but minimum case evaluations. The logic I use here is the one we use to solve SETS questions. Let me explain.

There are two couples. I don't want either couple to sit together. I would instead like to work with 'making them sit together'.

Would you agree that it is easy to find the number of arrangements in which both couples are sitting together? It is. We will work on it in a minute. Let me think ahead now.

How about 'finding the number of ways in which one couple sits together?' Sure we can find it but it will include those cases in which both couples are sitting together too. But we have already found the number of ways in which both couples sit together. We just subtract that number from this number and get the number of ways in which ONLY one couple sits together. Think of SETS here.

Let's do this now.

Number of arrangements in which both couples sit together: Say the couples are C1 and C2. I try and arrange the loner. He can take positions 1, 3 and 5. He can sit at 3 places. For each one of these positions, the couples can switch their places, C1 can switch places within themselves and C2 can switch places within themselves. So number of arrangements such that both couples are together are 3*2*2*2 = 24

Number of arrangements such that C1 is together: C1 acts as one group. Arrange 4 people/groups in 4! ways. C1 can switch places within themselves so number of arrangements = 4! * 2 = 48 But this 48 includes the number of arrangements in which both couples are sitting together. So number of arrangements such that ONLY C1 sits together = 48 - 24 = 24 Similarly, number of arrangements such that ONLY C2 sits together = 24

Total number of arrangements = 5! = 120 At least one couple sits together in 24 + 24 + 24 = 72 arrangements No couple sits together in 120 - 72 = 48 arrangements

Probability that no couple sits together = 2/5

(Ideally, you should see that 24 is 1/5th of 120 so you immediately arrive at 2/5) _________________

Re: very hard probability / combination (700+) helpppp [#permalink]
27 Oct 2011, 20:11

Expert's post

prashantbacchewar wrote:

Excellent approach Karishma. At what level in quants one can expect such a question in actual GMAT.

It is definitely a tough one. I would say closer to 750. The reason is that it can be time consuming and confusing if you do step by step case evaluations. _________________

Re: very hard probability / combination (700+) helpppp [#permalink]
17 Dec 2011, 21:26

Here is my approach:

P=Probability of neither sit together P1=probability of first couple sit together P2=probability of second couple sit together P3=probability of both couples sit together

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