J. If n is a positive even integer, then how many ways can n people be grouped into pairs? In the language of partitions, this problem is asking for the number of partitions of an n element set into subsets of cardinality two.

Suppose that in the first grade class from Exercise J of HW#4, the teacher selects one student from each of the six circular tables to sit at the front of the class in a line. How many seating arrangements of the entire class are possible? Note: the order of the students in the line at the front of the room matters, but we only care about the circular ordering of the students at each table. Also, the students at the front of the room bring their chairs, so each table has only four seats now.