Problem:
Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip?
Answer Choices:
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Solution:
Let , and denote the three different families in some order. Then the only possible arrangements are to have the second row be members of and the third row be members of , or to have the second row be members of and the third row be members of . Note that these are not the same, because in the first case one sibling pair occupy the right-most seat in the second row and the left-most seat in the third row, whereas in the second case this does not happen. (Having members of in the second row does not work because then the third row must be members of to avoid consecutive members of ; but in this case one of the siblings would be seated directly in front of the other sibling.) In each of these cases there are ways to assign the families to the letters and ways to position the boy and girl within the seats assigned to the families. Therefore the total number of seating arrangements is .
The problems on this page are the property of the MAA's American Mathematics Competitions