In how many ways can four married couples sit around a round table so that two women do not sit next to each other? How many arrangements are possible if we want to ensure that neither two spouses, nor two women, nor two men sit next to each other?
$1^{\circ}$. First, we seat the four ladies. These can be seated around the table in 3! ways. Then we place the gentlemen, so that one man is placed between each two ladies. Since the gentlemen can be seated in 44 ways, the total number of arrangements is

$$
3!4!=144
$$

$2^{\circ}$. Again, we first place the ladies. The number of their arrangements is 3! For example, $A, B, C, D$. We have already seen that if we want to ensure that no two women sit next to each other, then no two men can sit next to each other either. Then we place the gentlemen. Let's denote them by $a, b, c$, and $d$. Between $A$ and $B$, only $c$ or $d$ can be placed. If we allow $c$ to take the place, then $d$ can only be placed between $B$ and $C$, $a$ between $C$ and $D$, and $b$ between $D$ and $A$. If, however, $d$ is placed between $A$ and $B$, then the only possible arrangement is:

$$
A d B a C b D c
$$

Thus, we see that for each arrangement of the ladies, the gentlemen can be seated in two ways, so the total number of arrangements is

$$
2 \times 3!=12
$$

(Vilmos Erdös, Budapest.)