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 married couples, nor two women (nor two men) sit next to each other?
Let the husbands be denoted by $A, B, C, D$, and their wives respectively by $a, b, c, d$.

a) Since we are talking about a round table, $A$ can sit anywhere, and $B, C, D$ can take their seats in $3!=6$ different ways after him. Since no two women can sit next to each other, the 4 women have 4 available seats determined by the seating arrangement of the men. On these 4 seats, the 4 women can sit in $4!=24$ different ways, so the total number of possible seating arrangements is $6 \cdot 24=144$.

b) In the case of the seating order $A B C D$, the women can only sit in 2 different ways: cdab and dabc (see the 8th point of the third solution to problem 484, Volume VI, No. 5, p. 148). According to the 6 different seating arrangements of the husbands, the total number of possible seating arrangements is $6 \cdot 2=12$.

Imre Csiszár (Bp. I., Petőfi g. I. o. t.)