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If four married couples are arranged in a row, what is the probability that no husband sits next to his wife?

Would it be

$1- \frac{2(4!)}{8!}$?

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    How do you get $2(4!)$?2011-09-29
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    There are 2 ways the husband and wife can arrange themselves and 4! ways to order the the four husband/wife pairs, so there are 2*4! ways where the husband is next to the wife.2011-09-29
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    If **all four couples** are sitting next to each other, then you have $4!$ ways of arranging the couples, and $2$ ways of arranging **each** couple; that would be $2^4(4!)$ ways. But "all four couples are sitting next to each other" is not the complement of "no husband is sitting next to his wife"; you could have just *one* couple sitting together and the other three not.2011-09-29
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    Do you require that men and women alternate? That will impact the probability. Also, if you seat the couples as couples, the probability is zero.2011-09-29

3 Answers 3

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Let us first the label the couples as $A1,A2,A3$ and $A4$.Define the events $E1,E2,E3,E4$ as:

E1 = Event of A1 sitting together. E2 = Event of A2 sitting together. E3 = Event of A3 sitting together. E4 = Event of A4 sitting together. 

Hence,the probability $P\space(E1 \cup E2 \cup E3 \cup E4)$ will give the probability of the case where at least one of the four couples will sit together.

So, your required answer is $1-\space$$P\space(E1 \cup E2 \cup E3 \cup E4)$

PS:We can use the principle of mutual inclusion and exclusion to find $P\space(E1 \cup E2 \cup E3 \cup E4)$ and it should give $\frac{23}{35}$ (if I haven't made any mistake with calculation).

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    oops: sorry and never mind, I'm getting the same answer. I was comparing it to the answer to the actual question, not the computation of the latter probability. I think you're right, then.2011-09-29
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    @Arturo Magidin:Thanks,so according to my approach $1-\frac{23}{35} =\frac{12}{35}$ should be the correct answer.2011-09-29
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Assuming you seat the 8 individuals at random, one way of doing this is to use the inclusion exclusion principle and turn a number of couples into virtual individuals so they must sit next to each other to get $$1-\frac{ 2^1{4 \choose 1} 7! - 2^2{4 \choose 2} 6! + 2^3{4 \choose 3} 5! - 2^4{4 \choose 4} 4!}{8!}$$

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    Upvote for simplest solution. What would be your simplest solution for seating in a circle ?2015-05-31
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    @trueblueanil: Since there are (modulo rotation) $7!$ ways of arranging $8$ people around a table, I would say $1-\dfrac{ 2^1{4 \choose 1} 6! - 2^2{4 \choose 2} 5! + 2^3{4 \choose 3} 4! - 2^4{4 \choose 4} 3!}{7!}$. I have not checked this but for $2$ couples the equivalent statement seems to give $\frac13$ which seems to be correct.2015-05-31
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    This is great. So much simpler than Bogart and Doyle's solution to the relaxed menage problem. https://math.dartmouth.edu/~doyle/docs/menage/menage/menage.html May be a simpler solution to the menage problem also can be evolved starting from here.2015-06-01
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    @trueblueanil: I think my answer is pretty much the same as Bogart and Doyle, where my "turn a number of couples into virtual individuals" corresponds to their "non-overlapping dominos"2015-06-01
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    Of course, the paper is a classic, but by treating the circle as unnumbered, and not introducing an intermediate $d_k$, it becomes simpler for the relaxed menage problem.2015-06-02
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    Why you have done 7! ,6! 5! & 4!2016-10-17
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    @koolman: if you turn one couple into a virtual individual, you get $7$ individuals (real or virtual), who can be arranged $7!$ ways. Similarly with more couples turned into virtual individuals2016-10-17
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    @Henry but there will be 8 individuals (4 couples x 2)2016-10-17
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    @koolman: they are not quite fully individual if they are forced to sit next to each other2016-10-17
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    @Henry so why only 7!2016-10-17
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    We have 8 places for every individuals2016-10-17
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    @koolman: If you are asking about the round table, then you lose a degree of freedom for the rotational symmetry2016-10-17
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    @Henry no I am not asking of round table . I am asking for arrangement in a row.2016-10-17
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Let's count the complement. If at least one couple stands together, you have $\binom{4}{1} = 4$ ways of choosing the couple, $2$ ways of arranging them, and $7$ positions in which to place them (the first person in the couple cannot stand in the last place of the line), and the remaining 6 positions can be filled in $6!$ ways; this gives a total of $4\times 2\times 7\times 6! = 8!=40320$ ways.

However, this overcounts by counting arrangements in which more than one couple sits together twice. If at least two couples stand together, we have $4\times 3$ ways of choosing the first and second couple, $4$ ways of arranging them, 15 ways to stand them in the line in that order, and $4!$ of filling in the remaining 4 places, for a total of $4\times 3\times 4\times 15\times 4! = 17280$ ways.

If at least 3 couples stand together, we have $4\times 3\times 2$ ways of choosing the couples (in order), $8$ ways of arranging them amongst themselves, 10 ways of arranging them in the line, and $2$ ways of filling the remaining seats, for a total of $4\times 3\times 2\times 8\times 10\times 2= 3840$ ways.

If all four couples sit together, we have $4\times 3\times 2\times 1$ way of arranging the couples, and $2^4 = 16$ ways of arranging each couple, for a total of $384$ ways.

So, let's see: when we count "at least one couple stands together", we count the arrangements with exactly one couple together once, the arrangements with exactly two couples together twice; the arrangements with exactly three couples three times, and the arrangements with all four couples together four times.

When we could "at least two couples together", we count the arrangements with exactly two couples together once, the arrangements with exactly three couples together $\binom{3}{2}=3$ times; and the arrangements with all four couples together $\binom{4}{2}=6$ times.

When we could "at least three couples together", we count the arrangements with exactly three couples together once, and the arrangements with all four couples together $\binom{4}{3}=4$ times.

So if we take: $$(\text{at least one}) - (\text{at least two}) + (\text{at least three}) - (\text{all four})$$ we get the count exactly right.

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    Did not understand how did you get 15 and 10 in 2nd and 3rd calculation, can you please explain2018-08-14
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    @Hardikgupta: Did you see the date? It's been almost seven years since I wrote that.2018-08-14
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    I understand that, however I really liked your answer expcept that two values. it's ok, i'll try to decode it if I can (actually already tried hard and hence commented)2018-08-14
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    @Hardikgupta: Sigh... do you remember what you were thinking 7 years ago? Take the two couples. if the first couple stands in positions 1 and 2, then the second couple can stand in positions 3-4, 4-5, 5-6, 6-7, or 7-8. If first couple stands in 2-3, then second couple could be in 4-5, 5-6, 6-7, or 7-8. If first couple in 3-4, then 2nd in 5-6, 6-7, or 7-8; if first in 4-5, then second in 6-7 or 7-8; if first in 5-6, then second in 7-8. That's a total of 15 ways to arrange the two couples.2018-08-14