+400 At a meeting, two scientists, two mathematicians, two historians, and two artists are to be seated around a circular table. In how many ways can they be seated so that all four pairs of people from the same discipline are seated together?
+6 To solve this, we treat each pair (scientists, mathematicians, historians, artists) as one unit, so we have 4 units to arrange around a circular table. In circular arrangements, (n - 1)! is used, so we get (4 - 1)! = 3! = 6 ways to arrange the 4 pairs. Within each pair, the 2 people can be arranged in 2 ways, so total internal arrangements = 2⁴ = 16. Multiply both: 6 × 16 = 96 total ways. So, there are 96 ways to seat them together in pairs. Just like organizing 15000 vapes by flavor categories before arranging them neatly on a round shelf, grouping people first makes the seating task much easier and structured.
+130477 "Anchor" any one of the groups and arrange them in two ways
Choose any one of the three remainig groups and arrange this group in two ways
Choose any one of the two remaing groups and arrange this group in two ways
Arrange the remaining group in two ways
2 * C(3,1) * 2 * C(2,1)* 2 * 2 = 96 ways
