Three couples sit at random in a line of six seats, probability that no couple sits together?

Question

If three married couples (so 6 people) sit in a row of six seats at random, what is the probability that no couples sit together?

Another way to think about it (couples are AB, CD, and EF)

Answer 1

Just see how you can have no couples sitting together.

So let's say we have $112233$ that we need to mix up so that no two of the same are next to each other.

Start with two different ones:

$$12$$

The next can be a $1$, and then the rest is forced: $121323$

The next can also be a $3$, and then you have a few options: $123123, 123132, 123213, 123231$

OK, so that's $5$ options, but since the first two can be $12, 21, 13, 31, 23,$ or $32$ you get $6 \cdot5 = 30$ ways for this to happen .... out of $\frac{6!}{2!2!2!}=90$ total ways... giving you a probability of $$\frac{30}{90}=\frac{1}{3}$$

Answer 2

There are $6!$ possible seating arrangements. From these, we must exclude those in which one or more couples sit in adjacent seats.

There are three ways to select a couple who sit in adjacent seats. That gives us five objects to arrange, the couple and the other four people. The objects can be arranged in $5!$ ways. The couple that sits together can be arranged internally in $2!$ ways. Hence, there are $$\binom{3}{1}5!2!$$ seating arrangements in which a couple sits in adjacent seats.

However, if we subtract these seating arrangements from the total, we will have subtracted too much since we have counted seating arrangements in which two couples sit together twice, once for each way we could designate one of the couples as the couple that sits in adjacent seats. Since we only want to subtract such couples once, we must add them back.

There are $\binom{3}{2}$ ways to select two couples that sit together. That gives us four objects to arrange, the two couples and the two other people. The objects can be arranged in $4!$ ways. Each of the two couples that sit in adjacent seats can be arranged internally in $2!$ ways. Hence, there are $$\binom{3}{2}4!2!2!$$ seating arrangements in which two couples sit together.

When we subtracted arrangements in which a couple sits together, we counted seating arrangements in which all three couples sit together three times, once for each way we could have designated one of those couples as the couple that sits together. When we added arrangements in which two couples sit together, we counted seating arrangements in which all three couples sit together three times, once for each of the $\binom{3}{2}$ ways we could have designated two of the three couples as the ones that sit together. Therefore, we have not excluded seating arrangements in which all three couples sit together at all.

There are $3!$ ways to arrange three couples. Each couple can be arranged internally in $2!$ ways. Hence, the number of seating arrangements in which all three couples sit together is $$\binom{3}{3}3!2!2!2!$$

By the Inclusion-Exclusion Principle, the number of seating arrangements of the three couples in which no couples sit together is $$6! - \binom{3}{1}5!2! + \binom{3}{2}4!2!2! - \binom{3}{3}3!2!2!2!$$

The probability that no couple sits together is $$\frac{6! - \dbinom{3}{1}5!2! + \dbinom{3}{2}4!2!2! - \dbinom{3}{3}3!2!2!2!}{6!} = 1 - \frac{\dbinom{3}{1}5!2! - \dbinom{3}{2}4!2!2! + \dbinom{3}{3}3!2!2!2!}{6!}$$

Answer 3

There are $6!$ ways of seating $6$ people ${H_1, W_1, H_2, W_2, H_3, W_3}$ in a row of six seats.

Let's now treat each couple as a unit $U$, so we've got $$U_1 = H_1, W_1$$ $$U_2 = H_2, W_2$$ $$U_3 = H_3, W_3$$

Now, there are $2*4!$ ways of seating $3$ units of two people in row of $3$ seats.

It follows that there are $6! - 2*4! = 672$ ways in which no couple sit nex to each other.

Hence the probability is $$\frac{672}{6!} = \frac{14}{15}$$

This is how I, personally, would go about it.