+51 Four adults and four students are to be seated at a circular table. In how many different ways can they be seated if each adult must be next to two students? (Two seatings are considered the same if one can be rotated to form the other.)
+622 Since the arrangement is circular, any specific order will only be unique up to a rotation. So, choosing any person as the "starting point" and fixing them in a seat won't actually affect the number of possible arrangements.
Let's consider one adult, A. For A to be next to two students, there must be two students flanking A. There are 4 choices we can make for the first student to sit next to A.
Once we've chosen that student, there are 3 students left, and only one of them can sit on the other side of A. So, there are 4⋅1=4 ways to arrange the students such that A is flanked by two students.
Now, we have 3 adults left to seat. Since they each must be flanked by two students as well, we can repeat the same process for each adult, giving us 4⋅4⋅4=64 total arrangements.
