Solution
Couples Side by Side
Show the problem again
Three married couples sit around a round table with six seats. Two seatings are considered the same if one is a rotation of the other (mirror images count as different). In how many ways can they sit so that every couple occupies two adjacent seats? Give an integer.
Worked solution
The answer is 16. Glue each couple into a single block. Three blocks around a circle can be arranged in (3 − 1)! = 2 ways up to rotation. Within each block the two spouses can sit in either order, giving 2³ = 8 further choices. Since each person's spouse is a neighbour, the decomposition of any valid seating into blocks is unique, so nothing is double counted: 2 × 8 = 16.
Source: Classic counting exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.