# Here's a good problem

#### Jomo

##### Elite Member
Eight couples meet up at a restaurant. What is the probability that no one sits next to their mate? (assume they sit in a row)

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#### Denis

##### Senior Member
Eight couples meet up at a restaurant. What is the probability that no one sits next to their mate? (assume they sit in a row)
one turd

Staff member

#### Otis

##### Senior Member
… assume they sit in a row …
Are we supposed to also assume that the seats are assigned/chosen randomly?

If not, then is it a good exercise because it serves as practice for determining exercises that cannot be answered as given? :cool:

#### Denis

##### Senior Member
My 1/3 assumes completely at random.

Plus 1/3 also solution to all similar cases > 1 couple.

Do I get the "corner" again?

#### pka

##### Elite Member
Eight couples meet up at a restaurant. What is the probability that no one sits next to their mate? (assume they sit in a row)
Before retiring I loved to teach combinatorics at a simple level to in service teachers. They need renewal credits and a collection of problems giving "real world" applications. This was one I used to work on the concept of inclusion-exclusion. Say that the couples are named $$\displaystyle A,~B,~C~\&~D$$.
In order to find the number of ways that none of these four couples sits together we must subtract the number of ways at least on of the couples sit together from the total possible arrangements $$\displaystyle 8!$$. To find the of ways at least one couple is together we use inclusion/exclusion.
We add $$\displaystyle ||A||+||B||+||C||+||D||-||AB||-||AC||-||AD||-||BC||-||BD||-||CD||+||ABC||+||ABD||+||ACD||+||BCD||-||ABCD||$$
The couples taken on at a time minus two at a time plus three at a time and finally minus four at a time.
Now lets consider each of the four types.
$$\displaystyle ||A||=2\cdot 7!$$ is the number of ways couple $$\displaystyle A$$ can sit together. The 2 comes from the fact the wife can sit on either side of the husband.
Then that one unit along with the six others makes seven.

$$\displaystyle ||AB||=4\cdot 6!$$ is the number of ways couples $$\displaystyle A~\&~B$$ can each sit together, although the couples are not necessarily together.

$$\displaystyle ||ABC||=8\cdot 5!~\&~||ABCD||=16\cdot 4!$$

$$\displaystyle \displaystyle\sum\limits_{k = 0}^4 {{{( - 1)}^k}\left( {\begin{array}{*{20}{c}} 4\\ k \end{array}} \right)\left[ {{2^k}(8 - k)!} \right]}$$ is the number of ways that none of the couples sits together.

To see the probability that no couple is together SEE HERE

One can see how using a modified Mooore method, that can make a three hour night class.

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#### mmm4444bot

##### Super Moderator
Staff member
My 1/3 assumes completely at random …
I thought it was a riddle.

#### Romsek

##### Junior Member
My 1/3 assumes completely at random.

Plus 1/3 also solution to all similar cases > 1 couple.

Do I get the "corner" again?
$$\displaystyle \dfrac{12}{35} \Rightarrow \text{Pretty darn close to }\dfrac 1 3 \text{ but still }\Rightarrow \text{corner}$$

#### Denis

##### Senior Member
2 couples: 1,2,3,4 ; evens are the wives

Any combo containing one or more of pairs 12,21,34,43 are eliminated.

* = OK!
1234
1243
1324 *
1342
1423 *
1432

2134
2143
2314 *
2341
2413 *
2431

Similarly for 3??? and 4???

Clearly 2 out of 6 (or 8 out of 24)

Similarly did 3 couples and 4 couples (wrote program);
got similar results...so assumed it applied to all cases...

Where oh where did I goof?

#### pka

##### Elite Member
2 couples: 1,2,3,4 ;
Similarly did 3 couples and 4 couples (wrote program);
got similar results...so assumed it applied to all cases...
Where oh where did I goof?
Did you read the solution? This is a well known and used problem in counting courses.
The "goof" as you call it is your model.

#### Denis

##### Senior Member
OK OK...to the corner I go for 12/35 - 12/36 = 1/105 hours