Worked solution
Reserve the central front-row positions before placing the children
Four married couples attend a wedding dinner. One couple brings two children. Find the ways these ten people can sit around a table if
(i) there are no restrictions, mark
(ii) each couple must sit together. marks
They take a photo with the bride and bridegroom. The twelve people arrange themselves in two rows of six, with the bride and bridegroom together in the middle of the front row. How many arrangements also place both children in the front row? marks
[2010/NJC/II/7]
CJC · Review problems · Question 16
Answer
(i) 362880; (ii) 1920; photo 967680.
(i) Circular seating
Fix one of the ten distinct people as reference: .
(ii) Four couples and two children
Group each couple, giving six circular objects. The four couple blocks each have internal orders. Thus .
Photo method 1: choose the other front-row adults
The bride/groom occupy front-row positions 3 and 4, in orders. The two children must also be in that row; choose two of the original eight adults to complete it in ways. The four remaining front positions have orders for the children and chosen adults; the six back-row adults have orders. Total .
Photo method 2: place the children first
After reserving the central pair for bride/groom, choose two of the four other front-row positions for the children and order them: . Give bride/groom orders and arrange the original eight adults over the eight remaining positions in ways. Hence . The seating restriction on each original couple in(ii) is not a condition on this separate photo question.
Common mistakes
The bride and groom are additional people; the original eight adults are then distributed among the other eight unfilled photo positions.
Solution notes
All printed asks and marks retained, including the unnumbered final photo ask. Both source photo methods and complete factor interpretations retained.